traffic flow theory 2001

�

Traffic Flow Theory

A State-of-the-Art Report

Revised

2001

Organized by the Committee on Traffic Flow Theory and Characteristics (AHB45)

i

TABLE OF CONTENTS

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4

2. TRAFFIC STREAM CHARACTERISTICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12.1 Definitions and Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2

2.1.1 The Time-Space Diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22.1.2 Definitions of Some Traffic Stream Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22.1.3 Time-Mean and Space-Mean Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-42.1.4 Generalized Definitions of Traffic Stream Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-42.1.5 The Relation Between Density and Occupancy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-62..1.6 Three-Dimensional Representation of Vehicle Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8

2.2 Measurement Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-92.2.1 Measurement Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-92.2.2 Error Caused by the Mismatch Between Definitions and Usual Measurements .. . . . . . . . . . . 2-122.2.3 Importance of Location to the Nature of the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13

2.2.4 Selecting intervals from which to extract data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-142.3 Bivariate Models .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-15

2.3.1 Speed-Flow Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-162.3.2 Speed-Concentration Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-212.3.3 Flow-Concentration Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-26

2.4 Three-Dimensional Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-292.5 Summary and Llinks to Other Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-31

3. HUMAN FACTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

3.1.1 The Driving Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13.2 Discrete Driver Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3

3.2.1 Perception-Response Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3 3.3 Control Movement Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7

3.3.1 Braking Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7 3.3.2 Steering Response Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9

3.4 Response Distances and Times to Traffic Control Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9 3.4.1 Traffic Signal Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9 3.4.2 Sign Visibility and Legibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11 3.4.3 Real-Time Displays and Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-12 3.4.4 Reading Time Allowance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13

3.5 Response to Other Vehicle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13 3.5.1 The Vehicle Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13 3.5.2 The Vehicle Alongside . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14

3.6 Obstacle and Hazard Detection, Recognition, and Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-15 3.6.1 Obstacle and Hazard Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-15 3.6.2 Obstacle and Hazard Recognition and Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-15

3.7 Individual Differences in Driver Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-16 3.7.1 Gender . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-16 3.7.2 Age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-16 3.7.3 Driver Impairment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-17

ii

3.8 Continuous Driver Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-183.8.1 Steering Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-18

3.8.1.1 Human Transfer Function for Steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-183.8.1.2 Performance Characteristics Based on Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-19

3.9 Braking Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-203.9.1 Open-Loop Braking Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-203.9.2 Closed-Loop Braking Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-213.9.3 Less-Than-Maximum Braking Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-21

3.10 Speed and Acceleration Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-233.10.1 Steady-State Traffic Speed Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-233.10.2 Acceleration Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-23

3.11 Specific Maneuvers at the Guidance Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-23 3.11.1 Overtaking and Passing in the Traffic Stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-23

3.11.1.1 Overtaking and Passing Vehicles (4-Lane or 1-Way) . . . . . . . . . . . . . . . . . . . . . . . 3-233.11.1.2 Overtaking and Passing Vehicles (Opposing Traffic) . . . . . . . . . . . . . . . . . . . . . . . 3-24

3.12 Gap Acceptance and Merging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-243.12.1 Gap Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-243.12.2 Merging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-24

3.13 Stopping Sight Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-253.14 Intersection Sight Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-26

3.14.1 Case I: No Traffic Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-263.14.2 Case II: Yield Control for Secondary Roadway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-263.14.3 Case III: Stop Control on Secondary Roadway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-26

3.15 Other Driver Performance Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-273.15.1 Speed Limit Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-273.15.2 Distractors On/Near Roadway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-273.15.3 Real-Time Driver Information Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-28

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-28

4. CAR FOLLOWING MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-14.1 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-24.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6

4.2.1 Local Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-64.2.2 Asymptotic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9

4.2.1.1 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-104.2.1.2 Next-Nearest Vehicle Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13

4.3 Steady-State Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-144.4 Experiments And Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-20

4.4.1 Car Following Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-224.4.1.1 Analysis of Car Following Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-23

4.4.2 Macroscopic Observations: Single Lane Traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-324.5 Automated Car Following . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-384.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-38References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-39

5. CONTINUUM FLOW MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-15.1 Conservation and Traffic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-15.2 The Kinematic Wave Model of LWR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-6

5.2.1 The LWR Model and Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-6 5.2.2 The Riemann Problem and Entropy Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-7

5.2.3 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8

5.2.4 Extensions to the LWR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-95.2.5 Limitations of the LWR Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-11

5.3 High Order Continuum Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-135.3.1 Propagation of Traffic Sound Waves in Higher-Order Models . . . . . . . . . . . . . . . . . . . . . . . . . 5-155.3.2 Propagation of Shock and Expansion Waves .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-165.3.3 Traveling Waves, Instability and Roll Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-205.3.4 Summary and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-235 .4 Diffusive, Viscous and Stochastic Traffic Flow Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-245.4.1 Diffusive and Viscous Traffic Flow Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-245.4.2 Acceleration Noise and a Stochastic Flow Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-255.5 Numerical Approximations of Continuum Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-255.5.1 Finite Difference Methods for SOlving Inviscid Models .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-275.5.2 Finite Element Methods for Solving Viscous Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-305.5.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-335.5.3.1 Calibration of Model Parameters with Field Measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . 5-335.5.3.2 Multilane Traffic Flow Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-355.5.3.3 Traffic Flow on a Ring Road With a Bottleneck. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-35

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-45

6. MACROSCOPIC FLOW MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-16.1 Travel Time Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

6.1.1 General Traffic Characteristics as a Function of the Distance from the CBD . . . . . . . . . . . . . . . 6-26.1.2 Average Speed as a Function of Distance from the CBD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-3

6.2 General Network Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-66.2.1 Network Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-66.2.2 Speed and Flow Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-86.2.3 General Network Models Incorporating Network Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 6-116.2.4 Continuum Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-16

6.3 Two-Fluid Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-166.3.1 Two-Fluid Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-176.3.2 Two-Fluid Parameters: Influence of Driver Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-206.3.3 Two-Fluid Parameters: Influence of Network Features (Field Studies) . . . . . . . . . . . . . . . . . . 6-206.3.4 Two-Fluid Parameters: Estimation by Computer Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 6-226.3.5 Two-Fluid Parameters: Influence of Network Features (Simulation Studies) . . . . . . . . . . . . . 6-226.3.6 Two-Fluid Model: A Practical Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-23

6.4 Two-Fluid Model and Traffic Network Flow Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-236.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-25References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-29

7. TRAFFIC IMPACT MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17.1 Traffic and Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

7.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17.1.2 Flow and Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17.1.3 Logical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-27.1.4 Empirical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4

7.1.4.1 Kinds Of Study And Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-47.1.4.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-47.1.4.3 Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6

7.1.5 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7

iii

iv

7.2 Fuel Consumption Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-87.2.1 Factors Influencing Vehicular Fuel Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-87.2.2 Model Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-87.2.3 Urban Fuel Consumption Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-97.2.4 Highway Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-117.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-12

7.3 Air Quality Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-137.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-137.3.2 Air Quality Impacts of Transportation Control Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-137.3.3 Tailpipe Control Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-147.3.4 Highway Air Quality Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-15

7.3.4.1 UMTA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-157.3.4.2 CALINE-4 Dispersion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-157.3.4.3 Mobile Source Emission Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-167.3.4.4 MICRO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-187.3.4.5 The TRRL Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-19

7.3.5 Other Mobile Source Air Quality Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-20References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-20

8. UNSIGNALIZED INTERSECTION THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1

8.1.1 The Attributes of a Gap Acceptance Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-18.1.2 Interaction of Streams at Unsignalized Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-18.1.3 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1

8.2 Gap Acceptance Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-28.2.1 Usefulness of Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-28.2.2 Estimation of the Critical Gap Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-38.2.3 Distribution of Gap Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6

8.3 Headway Distributions Used in Gap Acceptance Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-68.3.1 Exponential Headways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-68.3.2 Displaced Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-78.3.3 Dichotomized Headway Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-78.3.4 Fitting the Different Headway Models to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-8

8.4 Interaction of Two Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-118.4.1 Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-118.4.2 Quality of Traffic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-168.4.3 Queue Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-198.4.4 Stop Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-228.4.5 Time Dependent Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-238.4.6 Reserve Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-268.4.7 Stochastic Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-27

8.5 Interaction of Two or More Streams in the Priority Road . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-288.5.1 The Benefit of Using a Multi-Lane Stream Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-28

8.6 Interaction of More than Two Streams of Different Ranking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-318.6.1 Hierarchy of Traffic Streams at a Two Way Stop Controlled Intersection . . . . . . . . . . . . . . . . 8-318.6.2 Capacity for Streams of Rank 3 and Rank 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-32

8.7 Shared Lane Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-358.7.1 Shared Lanes on the Minor Street . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-358.7.2 Shared Lanes on the Major Street . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-35

8.8 Two-Stage Gap Acceptance and Priority . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-368.9 All-Way Stop Controlled Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-37

8.9.1 Richardson’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-37

v

8.10 Empirical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-398.10.1 Kyte's Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-39

8.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-41References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-41

9. TRAFFIC FLOW AT SIGNALIZED INTERSECTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-19.2 Basic Concepts of Delay Models at Isolated Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-29.3 Steady-State Delay Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3

9.3.1 Exact Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-39.3.2 Approximate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-5

9.4 Time-Dependent Delay Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-109.5 Effect of Upstream Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-15

9.5.1 Platooning Effect On Signal Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-159.5.2 Filtering Effect on Signal Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-17

9.6 Theory of Actuated and Adaptive Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-199.6.1 Theoretically-Based Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-199.6.2 Approximate Delay Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-239.6.3 Adaptive Signal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-27

9.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-27References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-28

10. TRAFFIC SIMULATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-110.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-110.2 An Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-110.3 Car-Following . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-210.4 Random Number Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-210.5 Classification of Simulation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-310.6 Building Simulation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-510.7 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-510.8 Statistical Analysis of Simulation Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-17

10.8.1 Statistical Analysis for a Single System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1710.8.1.1 Fixed Sample-Size Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2010.8.1.2 Sequential Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-21

10.8.2 Alternative System Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2210.8.3 Variance Reduction Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2210.8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-23

10.9 Descriptions of Some Available Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2310.10 Looking to the Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-24References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-25

11. KINETIC THEORIES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1 11.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1

11.2 Status of the Prigogine-Herman Kinetic Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11-2 11.2.1 The Prigogine-Herman Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-2 11.2.2 Criticisms of the Prigogine-Herman Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-3 11.2.3 Accomplishments of the Prigogine-Herman Model. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-4 11.3 Other Kinetic Models

11.4 Continuum Models from Kinetic Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-611.5 Direct Solution of Kinetic Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-7References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..11-9

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-1

vi

LIST OF FIGURES

2. TRAFFIC STREAM CHARACTERISTICS

Figure 2.1Time-space Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2

Figure 2.2Trajectories in Time-space Region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5

Figure 2.3Trajectories of Vehicle Fronts and Rears. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7

Figure 2.4Three-dimensional representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-9

Figure 2.5Effect of measurement location on nature of data (modified from Hall, Hurdle, Banks 1992,and May 1990. . 2-17

Figure 2.6Generalized shape of speed-flow curve proposed by Hall, Hurdle and Banks (1992). . . . . . . . . . . . . . . . . . . 2-17

Figure 2.7Generalized shape of speed-flow curve proposed by Hall, Hurdle and Banks (1992).. . . . . . . . . . . . . . . . . . . 2-18

Figure 2.8Results from fitting polygon speed-flow curve to German data (Heidemann and Hotop). . . . . . . . . . . . . . . . . 2-18

Figure 2.9Data for 4-lane German Autobahns (2 lanes per direction), as reported by Stappert and Theis(1990). . . . . 2-20

Figure 2.10Greenshields' Speed-Flow Curve and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-20

Figure 2.11Greenshields' Speed-Density Graph and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-23

Figure 2.12Speed-Concentration Data from Merritt Parkway and Fitted Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-23

Figure 2.13Three Parts of Edie's Hypothesis for the Speed-Density Function, Fitted to Chicago Data . . . . . . . . . . . . . . . 2-25

Figure 2.14Greenshields' Speed-Flow Function Fitted to Chicago Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-28

Figure 2.15Four Days of Flow-Occupancy Data from Near Toronto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-28

Figure 2.16The Three-Dimensional Surface for Traffic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-30

Figure 2.17One Perspective on Three-dimensional Relationship (Gilchrist and Hall) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-30

Figure 2.18Second Perspective on Three-Dimensional Relationship (Gilchrist and Hall). . . . . . . . . . . . . . . . . . . . . . . . . . 2-32

Figure 2.19Catastrophe Theory Surface Showing Sketch of a Possible Freeway Function. . . . . . . . . . . . . . . . . . . . . . . . 2-32

vii

3. HUMAN FACTORS

Figure 3.1Generalized Block Diagram of the Car-Driver-Roadway System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2

Figure 3.2Lognormal Distribution of Perception-Reaction Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4

Figure 3.3A Model of Traffic Control Device Information Processing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10

Figure 3.4Looming as a Function of Distance from Object. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14

Figure 3.5Pursuit Tracking Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-19

Figure 3.6Typical Deceleration Profile for a Driver without Antiskid Braking System on a Dry Surface. . . . . . . . . . . . . . 3-22

Figure 3.7Typical Deceleration Profile for a Driver without Antiskid Braking System on a Wet Surface. . . . . . . . . . . . . 3-22

4. CAR FOLLOWING MODELS

Figure 4.1Schematic Diagram of Relative Speed Stimulus and a Weighing Function Versus Time . . . . . . . . . . . . . . . . . 4-4

Figure 4.1aBlock Diagram of Car-Following . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5

Figure 4.1bBlock Diagram of the Linear Car-Following Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5

Figure 4.2Detailed Motion of Two Cars Showing the Effect of a Fluctuation in the Acceleration of the Lead Car . . . . . . 4-8

Figure 4.3Changes in Car Spacings from an Original Constant Spacing Between Two Cars . . . . . . . . . . . . . . . . . . . . . . 4-9

Figure 4.4Regions of Asymptotic Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11

Figure 4.5Inter-Vehicle Spacings of a Platoon of Vehicles Versus Time for the Linear Car Following. . . . . . . . . . . . . . . 4-11

Figure 4.6Asymptotic Instability of a Platoon of Nine Cars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-12

Figure 4.7Envelope of Minimum Inter-Vehicle Spacing Versus Vehicle Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13

Figure 4.8Inter-Vehicle Spacings of an Eleven Vehicle Platoon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-14

Figure 4.9Speed (miles/hour) Versus Vehicle Concentration (vehicles/mile). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-17

Figure 4.10Normalized Flow Versus Normalized Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-17

Figure 4.11Speed Versus Vehicle Concentration(Equation 4.39) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-18

Figure 4.12Normalized Flow Versus Normalized Vehicle Concentration (Equation 4.40) . . . . . . . . . . . . . . . . . . . . . . . . . 4-18

Figure 4.13Normalized Flow Versus Normalized Concentration (Equations 4.51 and 4.52) . . . . . . . . . . . . . . . . . . . . . . 4-21

viii

Figure 4.14Normalized Flow versus Normalized Concentration Corresponding to the Steady-State Solution of Equations 4.51 and 4.52 for m=1 and Various Values of � . . . . . . . . . . . . . . . . . . . . 4-21

Figure 4.15Sensitivity Coefficient Versus the Reciprocal of the Average Vehicle Spacing. . . . . . . . . . . . . . . . . . . . . . . . . 4-24

Figure 4.16Gain Factor, �, Versus the Time Lag, T, for All of the Test Runs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-24

Figure 4.17Gain Factor, �, Versus the Reciprocal of the Average Spacing for Holland Tunnel Tests. . . . . . . . . . . . . . . . 4-25

Figure 4.18Gain Factor, � ,Versus the Reciprocal of the Average Spacing for Lincoln Tunnel Tests . . . . . . . . . . . . . . . . 4-26

Figure 4.19Sensitivity Coefficient, a ,Versus the Time Lag, T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-280,0

Figure 4.20Sensitivity Coefficient Versus the Reciprocal of the Average Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-29

Figure 4.21Sensitivity Coefficient Versus the Ratio of the Average Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-29

Figure 4.22Relative Speed Versus Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-31

Figure 4.23Relative Speed Thresholds Versus Inter-Vehicle Spacing for Various Values of the Observation Time. . . . . 4-32

Figure 4.24Speed Versus Vehicle Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-34

Figure 4.25Flow Versus Vehicle Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-34

Figure 4.26Speed Versus Vehicle Concentration (Comparison of Three Models) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-35

Figure 4.27Flow Versus Concentration for the Lincoln and Holland Tunnels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-36

Figure 4.28Average Speed Versus Concentration for the Ten-Bus Platoon Steady-State Test Runs . . . . . . . . . . . . . . . . 4-37

5. CONTINUUM FLOW MODELS

Figure 5.1Geometric Representation of Shocks, Sound Waves and Traffic Speeds in the k-q phase plane . . . . . . . . . . 5-4

Figure 5.2Field Representation of Shocks and Conservation of Flow. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 5-5

Figure 5.3A Shock Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8

Figure 5.4A Rarefaction Solution.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8

Figure 5.5Phase Transition Diagram in the Solution of Riemann Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20

Figure 5.6Roll Waves in the Moving Coordinate X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-22

Figure 5.7Traveling Waves and Shocks in the PW Modelic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-22

Figure 5.8Time-space Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-26

ix

Figure 5.9 The Kerner-Konhauser Model of Speed-Density and Flow-Density Relations. . . . . . . . . . . . . . . . . . . . . . . . . 5-36

Figure 5.10Initial Condition (114) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-36

Figure 5.11Solutinos of the Homogeneous LWR Model With Initial Condition in Figure 10 . . . . . . . . . . . . . . . . . . . . . . . . 5-37

Figure 5.12Initial Condition (116) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-38

Figure 5.13Solutions of the Inhomogeneous LWR Model With Initial Condition (116). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-39

Figure 5.14Solutions of the PW Model With Initial Condition (117). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-41

Figure 5.15Solutions of the PW Model With Initial Condition (118). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-42

Figure 5.16Comparison of the LWR Model and the PW Model on a Homogeneous Ring Road . . . . . . . . . . . . . . . . . . . 5-43

Figure 5.17Comparison of the LWR Model and the PW Model on an Inhomogeneous Ring Road. . . . . . . . . . . . . . . . . . 5-44

x

6. MACROSCOPIC FLOW MODELS

Figure 6.1Total Vehicle Distance Traveled Per Unit Area on Major Roads as a Function of the Distance from the Town Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2

Figure 6.2Grouped Data for Nottingham Showing Fitted (a) Power Curve, (b) Negative Exponential Curve, and (c) Lyman-Everall Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-4

Figure 6.3Complete Data Plot for Nottingham; Power Curve Fitted to the Grouped Data . . . . . . . . . . . . . . . . . . . . . . . . . 6-4

Figure 6.4Data from Individual Radial Routes in Nottingham, Best Fit Curve for Each Route is Shown . . . . . . . . . . . . . . 6-5

Figure 6.5Theoretical Capacity of Urban Street Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7

Figure 6.6Vehicles Entering the CBDs of Towns Compared with the CorrespondingTheoretical Capacities of the Road Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7

Figure 6.7Speeds and Flows in Central London, 1952-1966, Peak and Off-Peak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-8

Figure 6.8Speeds and Scaled Flows, 1952-1966 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9

Figure 6.9Estimated Speed-Flow Relations in Central London (Main Road Network) . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9

Figure 6.10Speed-Flow Relations in Inner and Outer Zones of Central Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-10

Figure 6.11Effect of Roadway Width on Relation Between Average (Journey) Speed and Flow in Typical Case . . . . . . 6-12

Figure 6.12Effect of Number of Intersections Per Mile on Relation Between Average (Journey) Speed and Flow in Typical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-12

Figure 6.13Effect of Capacity of Intersections on Relation Between Average (Journey) Speed and Flow in Typical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13

Figure 6.14Relationship Between Average (Journey) Speed and Number of Vehicles on Town Center Network . . . . . 6-13

Figure 6.15Relationship Between Average (Journey) Speed of Vehicles and Total Vehicle Mileage on Network . . . . . 6-14

Figure 6.16The �-Relationship for the Arterial Networks of London and Pittsburgh, in Absolute Values . . . . . . . . . . . . . 6-14

Figure 6.17The �-Relationship for the Arterial Networks of London and Pittsburgh, in Relative Values . . . . . . . . . . . . . 6-15

Figure 6.18The �-Map for London, in Relative Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-16

Figure 6.19Trip Time vs. Stop Time for the Non-Freeway Street Network of the Austin CBD . . . . . . . . . . . . . . . . . . . . . 6-18

Figure 6.20Trip Time vs. Stop Time Two-Fluid Model Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-19

Figure 6.21Trip Time vs. Stop Time Two-Fluid Model Trends Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-19

Figure 6.22Two-Fluid Trends for Aggressive, Normal, and Conservative Drivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-21

xi

Figure 6.23Simulation Results in a Closed CBD-Type Street Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-24

Figure 6.24Comparison of Model System 1 with Observed Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-26



7. TRAFFIC IMPACT MODELS

Figure 7.1Safety Performance Function and Accident Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2

Figure 7.2Shapes of Selected Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5

Figure 7.3Two Forms of the Model in Equation 7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6

Figure 7.4Fuel Consumption Data for a Ford Fairmont (6-Cyl.)Data Points represent both City and Highway Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-9

Figure 7.5Fuel Consumption Versus Trip Time per Unit Distance for a Number of Passenger Car Models. . . . . . . . . . 7-10

Figure 7.6Fuel Consumption Data and the Elemental Model Fit for Two Types of Passenger Cars . . . . . . . . . . . . . . . 7-10

Figure 7.7Constant-Speed Fuel Consumption per Unit Distance for the Melbourne University Test Car . . . . . . . . . . . . 7-12

8. UNSIGNALIZED INTERSECTION THEORY

Figure 8.1Data Used to Evaluate Critical Gaps and Move-Up Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3

Figure 8.2Regression Line Types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4

Figure 8.3Typical Values for the Proportion of Free Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-9

Figure 8.4Exponential and Displaced Exponential Curves (Low flows example). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-9

Figure 8.5Arterial Road Data and a Cowan (1975) Dichotomized Headway Distribution (Higher flows example). . . . . 8-10

Figure 8.6 Arterial Road Data and a Hyper-Erlang Dichotomized Headway Distribution (Higher Flow Example) . . . . . . 8-10

Figure 8.7Illustration of the Basic Queuing System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-12

Figure 8.8ComparisonRelation Between Capacity (q-m) and Priority Street Volume (q-p) . . . . . . . . . . . . . . . . . . . . . . . 8-14

Figure 8.9Comparison of Capacities for Different Types of Headway Distributions in the Main Street Traffic Flow . . . . 8-14

Figure 8.10The Effect of Changing � in Equation 8.31 and Tanner's Equation 8.36. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-15

Figure 8.11Probability of an Empty Queue: Comparison of Equations 8.50 and 8.52. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-18

xii

Figure 8.12Comparison of Some Delay Formulae. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-20

Figure 8.13Average Steady State Delay per Vehicle Calculated Using Different Headway Distributions. . . . . . . . . . . . . . 8-20

Figure 8.14Average Steady State Delay per Vehicle by Geometric Platoon Size Distribution and Different Mean Platoon Sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-21

Figure 8.1595-Percentile Queue Length Based on Equation 8.59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-22

Figure 8.16Approximate Threshold of the Length of Time Intervals For the DistinctionBetween Steady-State Conditions and Time Dependent Situations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-25

Figure 8.17The Co-ordinate Transform Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-25

Figure 8.18A Family of Curves Produced from the Co-Ordinate Transform Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . 8-27

Figure 8.19Average Delay, D, in Relation to Reserve Capacity R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-29

Figure 8.20Modified 'Single Lane' Distribution of Headways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-30

Figure 8.21Percentage Error in Estimating Adams' Delay Against the Major Stream Flow for a Modified Single Lane Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-31

Figure 8.22Traffic Streams And Their Level Of Ranking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-32

Figure 8.23Reduction Factor to Account for the Statistical Dependence Between Streams of Ranks 2 and 3. . . . . . . . . 8-33

Figure 8.24Minor Street Through Traffic (Movement 8) Crossing the Major Street in Two Phases. . . . . . . . . . . . . . . . . . 8-36

Figure 8.25Average Delay For Vehicles on the Northbound Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-40

9. TRAFFIC FLOW AT SIGNALIZED INTERSECTIONS

Figure 9.1 Deterministic Component of Delay Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2

Figure 9.2Queuing Process During One Signal Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3

Figure 9.3Percentage Relative Errors for Approximate Delay Models by Flow Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-9

Figure 9.4Relative Errors for Approximate Delay Models by Green to Cycle Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-9

Figure 9.5The Coordinate Transformation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-11

Figure 9.6Comparison of Delay Models Evaluated by Brilon and Wu (1990) with Moderate Peaking (z=0.50). . . . . . . . 9-14

Figure 9.7Comparison of Delay Models Evaluated by Brilon and Wu (1990) with High Peaking (z=0.70). . . . . . . . . . . 9-14

Figure 9.8Observations of Platoon Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-16

Figure 9.9HCM Progression Adjustment Factor vs Platoon Ratio Derived from TRANSYT-7F . . . . . . . . . . . . . . . . . . . 9-18

xiii

Figure 9.10Analysis of Random Delay with Respect to the Differential Capacity Factor (f) and Var/Mean Ratio of Arrivals (I)- Steady State Queuing Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-19

Figure 9.11Queue Development Over Time Under Fully-Actuated Intersection Control . . . . . . . . . . . . . . . . . . . . . . . . . . 9-21

Figure 9.12Example of a Fully-Actuated Two-Phase Timing Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-25

10. TRAFFIC SIMULATION

Figure 10.1Several Statistical Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-7

Figure 10.2Vehicle Positions During Lane-Change Maneuver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-8

Figure 10.3Structure Chart of Simulation Modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-9

Figure 10.4Comparison of Trajectories of Vehicles from Simulation Versus Field Data for Platoon 123. . . . . . . . . . . . 10-16

Figure 10.5Graphical Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-18

Figure 10.6Animation Snapshot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-19

11. KINETIC THEORIES

Figure 11.1 Dependence of the mean speed upon density normalized to jam density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-5Figure 11.2 Evolution of the flow, according to a diffusively corrected Lighthill-Whitham model. . . . . . . . . . . . . . . . . . . . . . 11-8

xiv

List of Tables

3. HUMAN FACTORS

Table 3.1Hooper-McGee Chaining Model of Perception-Response Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4

Table 3.2Brake PRT - Log Normal Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6

Table 3.3Summary of PRT to Emergence of Barrier or Obstacle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6

Table 3.4Percentile Estimates of PRT to an Unexpected Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7

Table 3.5Movement Time Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9

Table 3.6Visual Acuity and Letter Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11

Table 3.7Within Subject Variation for Sign Legibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-12

Table 3.8Object Detection Visual Angles (Daytime) (Minutes of Arc) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-15

Table 3.9Maneuver Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-19

Table 3.10Percentile Estimates of Steady State Unexpected Deceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-21

Table 3.11Percentile Estimates of Steady State Expected Deceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-21

Table 3.12Critical Gap Values for Unsignalized Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-25

Table 3.13PRTs at Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-27

4. CAR FOLLOWING MODELS

Table 4.1 Results from Car-Following Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-25

Table 4.2Comparison of the Maximum Correlations obtained for the Linear and Reciprocal Spacing Models for the Fourteen Lincoln Tunnel Test Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-27

Table 4.3Maximum Correlation Comparison for Nine Models, a , the Fourteen Lincoln Tunnel Test Runs. . . . . . . . . 4-28

� mTable 4.4

Results from Car Following Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-30Table 4.5

Macroscopic Flow Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-33Table 4.6

Parameter Comparison (Holland Tunnel Data) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-35

5. CONTINUUM MODELSTable 5.1 Oscillation Time and Magnitudes of Stop-and-go Traffic From German Measurement. . . . . . . . . . . . . . . . . . . 5-12

xv

7. TRAFFIC IMPACT MODELS

Table 7.1Federal Emission Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-14

Table 7.2Standard Input Values for the CALINE4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17

Table 7.3Graphical Screening Test Results for Existing Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-19


Table 8.1“A” Values for Equation 8.23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-8

Table 8.2Evaluation of Conflicting Rank Volume q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-34p


Table 9.1Maximum Relative Discrepancy between the Approximate Expressions and Ohno's Algorithm . . . . . . . . . . . 9-8

Table 9.2 Cycle Length Used For Delay Estimation for Fixed-Time and Actuated Signals Using Webster’s Formula . . 9-23

Table 9.3Calibration Results of the Steady-State Overflow Delay Parameter (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-26

10. TRAFFIC SIMULATION

Table 10.1Classification of the TRAF Family of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-4

Table 10.2Executive Routine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-9

Table 10.3Routine MOTIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-10

Table 10.4Routine CANLN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-11

Table 10. 5Routine CHKLC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-12

Table 10.6Routine SCORE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-13

Table 10.7Routine LCHNG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-14

Table 10.8Simulation Output Statistics: Measures of Effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-25

11. KINETIC THEORIES

Table 11.1 Status of various kinetic models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-6

Foreword - 1

FOREWORD

This publication is an update and expansion of Additional acknowledgment is made to Alberto Santiago,Transportation Research Board Special Report 165, Chief of State Programs at National Highway Institute of"Traffic Flow Theory," published in 1975. This updating the FHWA (formerly with Intelligent Systems andwas undertaken on recommendation of the Transportation Technology Division), without whose initiative andResearch Board's Committee A3A11 on Traffic Flow support, this report simply would not have been possible.Theory and Characteristics. The Federal Highway Thanks also go to Brenda Clark for the initial formattingAdministration (FHWA) funded a project to develop this of the report, Kathy Breeden for updating the graphicsreport via an Interagency Agreement with the Department and text and coordinating the effort with the authors, Philof Energy's Oak Ridge National Laboratory (ORNL). Wolff for the creation and management of the report’sThe project was carried out by ORNL under supervision web-site, and to Elaine Thompson for her projectof an Advisory Committee that, in addition to the three management assistance.co-editors, included the following prominent individuals:

Finally, we acknowledge the following individuals whoRichard Cunard, TRB Liaison RepresentativeDr. Henry Lieu, Federal Highway AdministrationDr. Hani Mahmassani, University of Texas at Austin

While the general philosophy and organization of theprevious two reports have been retained, the text hasbeen completely rewritten and two new chapters havebeen added. The primary reasons for doing such a majorrevision were to bring the material up-to-date; to includenew developments in traffic flow theory (e.g., networkmodels); to ensure consistency among chapters andtopics; and to emphasize the applications or practicalaspects of the theory. There are completely new chapterson human factors (Chapter 3) and network traffic models(Chapter 5).

To ensure the highest degree of reliability, accuracy, andquality in the content of this report, the collaboration of alarge number of experts was enlisted, and this reportpresents their cooperative efforts. We believe that aserious effort has been made by the contributing authorsin this report to present theory and information that willhave lasting value. Our appreciation is extended to themany authors for their commendable efforts in writing Texas A&M Universitythis update, willingly sharing their valuable time,knowledge, and their cooperative efforts throughout the Dr. Ajay K. Rathiproject.

We would also like to acknowledge the time spent by themembers of the Advisory Committee in providingguidance and direction on the style of the report andfor their reviews of the many drafts of the report.

read and reviewed part or all of the manuscript andcontributed valuable suggestions: Rahmi Akcelik, RahimBenekohal, David Boyce, Micheal Brackstone, WernerBrilon, Christine Buisson, Ennio Cascetta, MichaelCassidy, Avishai Ceder, Arun Chatterjee, Ken Courage,Ray Derr, Mike Florian, Fred Hall, Benjamin Heydecker,Ben Hurdle, Shinya Kikuchi, Helmut “Bill” Knee, HarisKoutsopoulos, Jean-Baptiste Lesort, John Leonard II,Fred Mannering, William McShane, Kyriacos Mouskos,Panos Prevedourous, Vladimir Protopopescu, Bin Ran,Tom Rockwell, Mitsuru Saito, and Natacha Thomas.

We believe that this present publication meets itsobjective of synthesizing and reporting, in a singledocument, the present state of knowledge or lack thereofin traffic flow theory. It is sincerely hoped that this reportwill be useful to the graduate students, researchers andpractitioners, and others in the transportation profession.

Editors: Dr. Nathan GartnerUniversity of Massachusetts - Lowell

Dr. Carroll J. Messer

Oak Ridge National LaboratoryProject Leader.

INTRODUCTIONBY NATHAN H. GARTNER1

CARROLL MESSER2

AJAY K. RATHI3

Professor, Department of Civil Engineering, University of Massachusetts at Lowell, 1 University Avenue, Lowell, MA 01854.1

Professor, Department of Civil Engineering, Texas A&M University, TTI Civil Engineering Building, Suite 304C, College Station,2

TX 77843-3135. Senior R&D Program Manager and Group Leader, ITS Research, Center for Transportation Analysis, Oak Ridge National Laboratory,3

P.O. Box 2008, Oak Ridge, TN 37831-6207.

1. INTRODUCTION

1 - 1

1.INTRODUCTION

It is hardly necessary to emphasize the importance of By 1959 traffic flow theory had developed to the point where ittransportation in our lives. In the United States, we spend about appeared desirable to hold an international symposium. The20 percent of Gross National Product (GNP) on transportation, First International Symposium on The Theory of Traffic Flowof which about 85 percent is spent on highway transportation was held at the General Motors Research Laboratories in(passenger and freight). We own and operate 150 million Warren, Michigan in December 1959 (Herman 1961). This wasautomobiles and an additional 50 million trucks, bringing car the first of what has become a series of triennial symposia onownership to 56 per hundred population (highest in the world). The Theory of Traffic flow and Transportation. The most recentThese vehicles are driven an average of 10,000 miles per year in this series, the 12th symposium was held in Berkeley,for passenger cars and 50,000 miles per year for trucks on a California in 1993 (Daganzo 1993). A glance through thehighway system that comprises more than 4 million miles. The proceedings of these symposia will provide the reader with aindices in other countries may be somewhat different, but the good indication of the tremendous developments in theimportance of the transportation system, and especially the understanding and the treatment of traffic flow processes in thehighway component of it, is just the same or even greater. While past 40 years. Since that time numerous other symposia andcar ownership in some countries may be lower, the available specialty conferences are being held on a regular basis dealinghighway network is also smaller leading to similar or more with a variety of traffic related topics. The field of traffic flowsevere congestion problems. theory and transportation has become too diffuse to be covered

Traffic flow theories seek to describe in a precise mathematical flow theory, while better understood and more easilyway the interactions between the vehicles and their operators characterized through advanced computation technology, are just(the mobile components) and the infrastructure (the immobilecomponent). The latter consists of the highway system and all itsoperational elements: control devices, signage, markings, etc.As such, these theories are an indispensable construct for allmodels and tools that are being used in the design and operationof streets and highways. The scientific study of traffic flow hadits beginnings in the 1930’s with the application of probabilitytheory to the description of road traffic (Adams 1936) and thepioneering studies conducted by Bruce D. Greenshields at theYale Bureau of Highway Traffic; the study of models relatingvolume and speed (Greenshields 1935) and the investigation ofperformance of traffic at intersections (Greenshields 1947).After WWII, with the tremendous increase in use of automobilesand the expansion of the highway system, there was also a surgein the study of traffic characteristics and the development oftraffic flow theories. The 1950’s saw theoretical developmentsbased on a variety of approaches, such as car-following, trafficwave theory (hydrodynamic analogy) and queuing theory. Someof the seminal works of that period include the works byReuschel (1950a; 1950b; 1950c), Wardrop (1952), Pipes(1953), Lighthill and Whitham (1955), Newell (1955), Webster(1957), Edie and Foote (1958), Chandler et al. (1958) and otherpapers by Herman et al. (see Herman 1992).

by any single type of meeting. Yet, the fundamentals of traffic

as important today as they were in the early days. They form thefoundation for all the theories, techniques and procedures thatare being applied in the design, operation, and development ofadvanced transportation systems.

It is the objective of this monograph to provide an updatedsurvey of the most important models and theories thatcharacterize the flow of highway traffic in its many facets. Thismonograph follows in the tracks of two previous works that weresponsored by the Committee on Theory of Traffic Flow of theTransportation Research Board (TRB) and its predecessor theHighway Research Board (HRB). The first monograph, whichwas published as HRB Special Report 79 in 1964, consisted ofselected chapters in the then fledgling Traffic Science each ofwhich was written by a different author (Gerlough and Capelle1964). The contents included:

Chapter 1. Part I: Hydrodynamic Approaches, by L. A. Pipes.Part II: On Kinematic Waves; A Theory of Traffic Flow on LongCrowded Roads, by M. J. Lighthill and G. B. Whitham.

Chapter 2. Car Following and Acceleration Noise, by E. W.Montroll and R. B. Potts.

1. INTRODUCTION

1 - 2

Chapter 3. Queuing Theory Approaches, by D. E. Cleveland their entirety and include the latest research and information inand D. G. Capelle. their respective areas. Chapter 2 presents the various models

Chapter 4. Simulation of Traffic Flow, by D. L. Gerlough. the traffic stream variables: speed, flow, and concentration.

Chapter 5. Some Experiments and Applications, by R. S. Foote. flow, primarily on freeways or expressways. The chapter

A complete rewriting of the monograph was done by Gerlough since to as large extent development of the first depends on theand Huber (1975) and was published as TRB Special Report latter.165 in 1975. It consisted of nine chapters, as follows:

Chapter 1. Introduction. aspects of the human element in the context of the person-

Chapter 2. Measurement of Flow, Speed, and Concentration. first discrete components of performance, including: perception-

Chapter 3. Statistical Distributions of Traffic Characteristics. control devices, to the movement of other vehicles, to hazards in

Chapter 4. Traffic Stream Models. in performance. Next, the kind of control performance that

Chapter 5. Driver Information Processing Characteristics. control functions -- is described. Applications of open-loop and

Chapter 6. Car Following and Acceleration Noise. keeping, car following, overtaking, gap acceptance, lane

Chapter 7. Hydrodynamic and Kinematic Models of Traffic. the chapter, a few other performance aspects of the driver-

Chapter 8. Queuing Models (including Delays at distractions on the highway, and responses to real-time driverIntersections). information. The most obvious application of human factors is

Chapter 9. Simulation of Traffic Flow. following models examine the manner in which individual

This volume is now out of print and in 1987 the TRB Committee are developed from a stimulus-response relationship, where theon Traffic Flow Theory and Characteristics recommended that response of successive drivers in the traffic stream is toa new monograph be prepared as a joint effort of committee accelerate or decelerate in proportion to the magnitude of themembers and other authors. While many of the basic theories stimulus at time t after a time lag T. Car following models formmay not have changed much, it was felt that there were a bridge between the microscopic behavior of individual vehiclessignificant developments to merit writing of a new version of the and the macroscopic characteristics of a single-lane trafficmonograph. The committee prepared a new outline which stream with its corresponding flow and stability properties.formed the basis for this monograph. After the outline wasagreed upon, the Federal Highway Administration supported this Chapter 5 deals with Continuous Flow Models. Because trafficeffort through an interagency agreement with the Oak Ridge involves flows, concentrations, and speeds, there is a naturalNational Laboratory. An Editorial Committee was appointed, tendency to attempt to describe traffic in terms of fluid behavior.consisting of N. H. Gartner, C. J. Messer, and A. K. Rathi, which Car following models recognize that traffic is made up ofwas charged with the editorial duties of the preparation of the discrete particles and it is the interactions between thesemanuscripts for the different chapters. particles that have been developed for fluids (i.e., continuum

The first five chapters follow similarly titled chapters in the of the traffic stream rather than with the interactions between theprevious monograph; however, they all have been rewritten in particles. In the fluid flow analogy, the traffic stream is treated

that have been developed to characterize the relationships among

Most of the relationships are concerned with uninterrupted traffic

stresses the link between theory and measurement capability,

Chapter 3, on Human Factors, discusses salient performance

machine system, i.e. the motor vehicle. The chapter describes

reaction time, control movement time, responses to traffic

the roadway, and how different segments of the population differ

underlies steering, braking, and speed control -- the primary

closed-loop vehicle control to specific maneuvers such as lane

closures, and sight distances are also described. To round out

vehicle system are covered, such as speed limit changes,

in the development of Car Following Models (Chapter 4). Car

vehicles (and their drivers) follow one another. In general, they

models) is concerned more with the over all statistical behavior

1. INTRODUCTION

1 - 3

as a one dimensional compressible fluid. This leads to two basic intersections among two or more streams and providesassumptions: (i) traffic flow is conserved, which leads to the calculations of capacities and quality of traffic operations basedconservation or continuity equation, and (ii) there is a one-to-one on queuing modeling. relationship between speed and density or between flow anddensity. The simple continuum model consists of the Traffic Flow at Signalized Intersections is discussed in Chapterconservation equation and the equation of state (speed-density or 9. The statistical theory of traffic flow is presented, in order toflow-density relationship). If these equations are solved together provide estimates of delays and queues at isolated intersections,with the basic traffic flow equation (flow equals density times including the effect of upstream traffic signals. This leads to thespeed) we can obtain the speed, flow and density at any time and discussion of traffic bunching, dispersion and coordination atpoint of the roadway. By knowing these basic traffic flow traffic signals. The fluid (shock-wave) approach to trafficvariables, we know the state of the traffic system and can derive signal analysis is not covered in this chapter; it is treated tomeasures of effectiveness, such as delays, stops, travel time, total some extent in Chapter 5. Both pretimed and actuated signaltravel and other measures that allow the analysts to evaluate how control theory are presented in some detail. Further, delaywell the traffic system is performing. In this chapter, both models that are founded on steady-state queue theory as well assimple and high order models are presented along with analytical those using the so-called coordinate transform method areand numerical methods for their implementation. covered. Adaptive signal control is discussed only in a

Chapter 6, on Macroscopic Flow Models, discards the development of optimal signal control strategies, which ismicroscopic view of traffic in terms of individual vehicles or outside the scope of this chapter.individual system components (such as links or intersections)and adopts instead a macroscopic view of traffic in a network. The last chapter, Chapter 10, is on Traffic Simulation.A variety of models are presented together with empirical Simulation modeling is an increasingly popular and effective toolevidence of their applicability. Variables that are being for analyzing a wide variety of dynamical problems which areconsidered, for example, include the traffic intensity (the distance not amenable to study by other means. These problems aretravelled per unit area), the road density (the length or area of usually associated with complex processes which can not readilyroads per unit area of city), and the weighted space mean speed. be described in analytical terms. To provide an adequate testThe development of such models extends traffic flow theory into bed, the simulation model must reflect with fidelity the actualthe network level and can provide traffic engineers with the traffic flow process. This chapter describes the traffic modelsmeans to evaluate system-wide control strategies in urban areas. that are embedded in simulation packages and the proceduresFurthermore, the quality of service provided to motorists can be that are being used for conducting simulation experiments. monitored to assess the city's ability to manage growth. Networkperformance models could also be used to compare traffic Consideration was also given to the addition of a new chapter onconditions among different cities in order to determine the Traffic Assignment Models. Traffic assignment is the processallocation of resources for transportation system improvements. of predicting how a given set of origin-destination (OD) tripChapter 7 addresses Traffic Impact Models, specifically, the demands will manifest themselves onto a transportation networkfollowing models are being discussed: Traffic and Safety, Fuel of links and nodes in terms of flows and queues. It has majorConsumption Models, and Air Quality Models. applications in both transportation planning models and in

Chapter 8 is on Unsignalized Intersection Theory. Unsignalized Intelligent Transportation Systems (ITS). Generally, theintersections give no positive indication or control to the driver. assignment process consists of a macroscopic simulation of theThe driver alone must decide when it is safe to enter the behavior of travelers in a network of transportation facilities. Atintersection, typically, he looks for a safe opportunity or "gap" in the same time it reflects the interconnection between thethe conflicting traffic. This model of driver behavior is called microscopic models of traffic behavior that are discussed in this"gap acceptance." At unsignalized intersections the driver must monograph and the overall distribution of traffic demandsalso respect the priority of other drivers. This chapter discusses throughout the network. This is expressed by link cost functionsin detail the gap acceptance theory and the headway distributions that serve as a basis for any assignment or route choice process.used in gap acceptance calculations. It also discusses the After much deliberation by the editorial and advisory committees

qualitative manner since this topic pertains primarily to the

dynamic traffic management models which are the bedrock of

1. INTRODUCTION

1 - 4

it was decided that the subject cannot be presented adequately in and Iida (1997). This is a lively research area and newa short chapter within this monograph. It would be better served publications abound.by a dedicated monograph of its own, or by reference to theextensive literature in this area. Early references include the Research and developments in transportation systems and,seminal works of Wardrop (1952), and Beckmann, McGuire and concomitantly, in the theories that accompany them proceed atWinsten (1956). Later publications include books by Potts and a furious pace. Undoubtedly, by the time this monograph isOliver (1972), Florian (1976), Newell (1980), and Sheffi printed, distributed, and read, numerous new developments will(1985). Recent publications, which reflect modern approaches have occurred. Nevertheless, the fundamental theories will notto equilibrium assignment and to dynamic traffic assignment, have changed and we trust that this work will provide a usefulinclude books by Patriksson (1994), Ran and Boyce (1994), source of information for both newcomers to the field andGartner and Improta (1995), Florian and Hearn (1995), and Bell experienced workers.

References

Adams, W. F. (1936). Road Traffic Considered as a Random Greenshields, B. D. (1935). A Study in Highway Capacity.Series, J. Inst. Civil Engineers, 4, pp. 121-130, U.K.

Beckmann, M., C.B. McGuire and C.B. Winsten (1956). Studies in the Economics of Transportation. YaleUniversity Press, New Haven.

Bell, M.G.H. and Y. Iida (1997). Transportation NetworkAnalysis. John Wiley & Sons.

Chandler, R. E., R. Herman, and E. W. Montroll, (1958). Traffic Dynamics: Studies in Car Following, Opns. Res. 6, pp. 165-183.

Daganzo, C. F., Editor (1993). Transportation and TrafficTheory. Proceedings, 12th Intl. Symposium. ElsevierScience Publishers.

Edie, L. C. and R. S. Foote, (1958). Traffic Flow in Tunnels,Proc. Highway Research Board, 37, pp. 334-344.

Florian, M.A., Editor (1976). Traffic Equilibrium Methods.Lecture Notes in Economics and Mathematical Systems, Flowing Highway Traffic. Operations Research 3, Springer-Verlag. pp. 176-186.

Florian, M. and D. Hearn (1995). Network Equilibrium Newell, G. F. (1980). Traffic Flow on TransportationModels and Algorithms. Chapter 6 in Network Routing Networks. The MIT Press, Cambridge, Massachusetts.(M.O. Ball et al., Editors), Handbooks in OR & MS, Vol.8, Elsevier Science.

Gartner, N.H. and G. Improta, Editors (1995). Urban TrafficNetworks; Dynamic Flow Modeling and Control.Springer-Verlag.

Gerlough, D. L. and D. G. Capelle, Editors (1964). AnIntroduction to Traffic Flow Theory. Special Report 79.Highway Research Board, Washington, D.C.

Gerlough, D. L. and M. J. Huber, (1975). Traffic Flow Theory- A Monograph. Special Report 165, TransportationResearch Board.

Highway Research Board, Proceedings, Vol. 14, p. 458.Greenshields, B. D., D. Schapiro, and E. L. Erickson, (1947).

Traffic Performance at Urban Intersections. Bureau ofHighway Traffic, Technical Report No. 1. Yale UniversityPress, New Haven, CT.

Herman, R., Editor (1961). Theory of Traffic Flow. ElsevierScience Publishers.

Herman, R., (1992). Technology, Human Interaction, andComplexity: Reflections on Vehicular Traffic Science.Operations Research, 40(2), pp. 199-212.

Lighthill, M. J. and G. B. Whitham, (1955). On KinematicWaves: II. A Theory of Traffic Flow on Long CrowdedRoads. Proceedings of the Royal Society: A229, pp. 317-347, London.

Newell, G. F. (1955). Mathematical Models for Freely

Patriksson, M. (1994). The Traffic Assignment Problem;Models and Methods. VSP BV, Utrecht, The Netherlands.

Pipes, L. A. (1953). An Operational Analysis of TrafficDynamics. J. Appl. Phys., 24(3), pp. 274-281.

Potts, R.B. and R.M. Oliver (1972). Flows in TransportationNetworks. Academic Press.

Ran, B. and D. E. Boyce (1994). Dynamic Urban Transportation Network Models; Theory and Implicationsfor Intelligent Vehicle-Highway Systems. Lecture Notes inEconomics and Mathematical Systems, Springer-Verlag.

1. INTRODUCTION

1 - 5

Reuschel, A. (1950a). Fahrzeugbewegungen in der Kolonne. Wardrop, J. G. (1952). Some Theoretical Aspects of RoadOesterreichisches Ingenieur-Aarchiv 4, No. 3/4, pp. 193-215.

Reuschel, A. (1950b and 1950c). Fahrzeugbewegungen in derKolonne bei gleichformig beschleunigtem oder verzogertem Technical Paper No. 39. Road Research Laboratory,Leitfahrzeug. Zeitschrift des Oesterreichischen Ingenieur-und Architekten- Vereines 95, No. 7/8 59-62, No. 9/10 pp.73-77.

Sheffi, Y. (1985). Urban Transportation Networks;Equilibrium Analysis with Mathematical ProgrammingMethods. Prentice-Hall.

Traffic Research. Proceedings of the Institution of CivilEngineers, Part II, 1(2), pp. 325-362, U.K.

Webster, F. V. (1958). Traffic Signal Settings. Road Research

London, U.K.

TRAFFIC STREAM CHARACTERISTICSBY FRED L. HALL4

Professor, McMaster University, Department of Civil Engineering and Department of Geography, 1280 Main Street West,4

Hamilton, Ontario, Canada L8S 4L7.

CHAPTER 2 - Frequently used Symbols

k density of a traffic stream in a specified length of road

L length of vehicles of uniform length

c constant of proportionality between occupancy andkdensity, under certain simplifying assumptions

k the (average) density of vehicles in substream Ii

q the average rate of flow of vehicles in substream Ii

� average speed of a set of vehicles

A A(x,t) the cumulative vehicle arrival function overspace and time

k jam density, i.e. the density when traffic is so heavy thatjit is at a complete standstill

u free-flow speed, i.e. the speed when there are nofconstraints placed on a driver by other vehicles on theroad

Fourth draft of revised chapter 2 2001 October 14 Chapter 2: Traffic Stream Characteristics

Author’s note: This material has benefited greatly from the assistance of Michael Cassidy, of the University of California at Berkeley. He declined the offer to be listed as a co-author of the chapter, although that would certainly have been warranted. With his permission, and that of the publisher, several segments of this material have been reproduced directly from his chapter, “Traffic Flow and Capacity”, which appears as Chapter 6 in Handbook of transportation science, edited by Randolph W. Hall, and published by Kluwer Academic Publishers in 1999. That material appears in italics below, in section 2.1. The numbering of Figures and equations has been altered from his numbers to correspond to numbering within this chapter. In this chapter, properties that describe highway traffic are introduced, such as flow, density, and average vehicle speed; issues surrounding their measurements are discussed; and various models that have been proposed for describing relationships among these properties are presented. Most of the work dealing with these relationships has been concerned with uninterrupted traffic flow, primarily on freeways or expressways, but the general relationships will apply to all kinds of traffic flow. This chapter starts with a section on definitions of key variables and terms. Because of the importance of measurement capability to theory development, the second section deals with measurement issues, including historical developments in measurement procedures, and criteria for selecting good measurement characteristics. The third section discusses a number of the bivariate models that have been proposed in the past to relate key variables. That is followed by a short section on attempts to deal simultaneously with the three key variables. The final, summary section provides links to a number of the other chapters in this monograph. 2.1 Definitions and terms This section provides definitions of some properties commonly used to characterize traffic streams, together with some generalized definitions that preserve useful relations among these properties. Before turning to the definitions, however, a useful graphical tool is introduced, namely trajectories plotted on a time-space diagram. This is followed by definitions of traffic stream properties, time-mean and space-mean properties, and the generalized definitions. Following that is a discussion of the relationship between density, which is often used in the generalized definitions, and occupancy, which is measured by many freeway systems. The final topic covered is three-dimensional representations of traffic streams. The discussion in this section follows very closely or repeats some of that in the chapter by Cassidy (1999), which in turn credits notes composed by Newell (unpublished) and a textbook written by Daganzo (1997).


2-1

2.1.1 The Time-Space Diagram. The Time-Space Diagram. Objects are commonly constrained to move along a one-dimensional guideway, be it, for example, a highway lane, walkway, conveyor belt, charted course or flight path. Thus, the relevant aspects of their motion can often be described in cartesian coordinates of time, t, and space, x. Figure 2-1 illustrates the trajectories of some objects traversing a facility of length L during time interval T; these objects may be vehicles, pedestrians or cargo. Each trajectory is assigned an integer label in the ascending order that the object would be seen by a stationary observer. If one object overtakes another, their trajectories may exchange labels, as shown for the fourth and fifth trajectories in the figure. Thus, the �th trajectory describes the location of a reference point (e.g. the front end) of object � as a function of time t, x� (t).

Figure 2-1. Time-space diagram.

The characteristic geometries of trajectories on a time-space diagram describe the motion of

objects in detail. These diagrams thus offer the most complete way of displaying the observations that may have actually been measured along a facility. As a practical matter, however, one is not likely to collect all the data needed to construct trajectories. Rather, time-space diagrams derive their (considerable) value by providing a means to highlight the key features of a traffic stream using only coarsely approximated data or hypothetical data from “thought experiments.”

2.1.2 Definitions of Some Traffic Stream Properties. It is evident from Figure 2-1 that the slope of the �th trajectory is object �’s instantaneous velocity, v� (t), i.e., v (t) � dx (t)/dt , (2.1) and that the curvature is its acceleration. Further, there exist observable properties of a traffic stream that relate to the times that objects pass a fixed location, such as location x1, for example.


2-2

These properties are described with trajectories that cross a horizontal line drawn through the time-space diagram at x1.

Referring to Figure 2-1, the headway of some ith object at x1, hi(x1), is the difference between the

arrival times of i and i-1 at x1, i.e., hi(x1 ) � ti(x1 ) - ti-1(x1 ) (2.2) Flow at x1 is m, the number of objects passing x1, divided by the observation interval T,

q(T, x1 ) � m/T. (2.3) For observation intervals containing large m,

� ��

m

ii Txh

11)(

(2.4)

and thus, ,

)(1

)(11),(

1

11

1 xhxhm

x Tqm

ii

��

�

�

(2.5)

i.e., flow is the reciprocal of the average headway. Analogously, some properties relate to the locations of objects at a fixed time, as observed, for example, from an aerial photograph. These properties may be described with trajectories that cross a vertical line in the t-x plane. For example, the spacing of object j at some time t1, sj(t1 ), is the distance separating j from the next downstream object; i.e.,

sj(t1 ) � xj-1(t1 ) - xj(t1 ). (2.6) Density at instant t1 is n, the number of objects on a facility at that time, divided by L, the facility’s physical length; i.e.,

k(L, t1 ) � n /L. (2.7) If the L contains large n,

(2.8) � ��

n

jj Lts

11)(

and

,

)(1

)(11),(

1

11

1 tstsn

t Lkn

jj

��

�

�

(2.9)

giving a relation between density and the average spacing parallel to that of flow and the average headway.


2-3

2.1.3 Time-Mean and Space-Mean Properties. For an object’s attribute �, where � might be its velocity, physical length, number of occupants, etc., one can define an average of the m objects passing some fixed location x1 over observation interval T,

,)(1),(

111 � ��

�

m

ii x

mx T (2.10)

i.e., a time-mean of attribute �. If � is headway, for example, �(T, x1) is the average headway or the reciprocal of the flow. Conversely, the space-mean of attribute � at some time t1, �(L, t1), is obtained from the observations taken at that time over a segment of length L, i.e.,

.)(1),(1

11 � ��

n

jj t

ntL

(2.11)

If, for example, � is spacing, �(L, t1) is the average spacing or the reciprocal of the density. For any attribute �, there is no obvious relation between its time and space means. The reader may confirm this (using the example of � as velocity) by envisioning a rectangular time-space region L � T traversed by vehicles of two classes, fast and slow, which do not interact. For each class, the trajectories are parallel, equidistant and of constant slope; such conditions are said to be stationary. The fraction of fast vehicles distributed over L as seen on an aerial photograph taken at some instant within T will be smaller than the fraction of fast vehicles crossing some fixed point along L during the interval T. This is because the fast vehicles spend less time in the region than do the slow ones. Analogously, one might envision a closed loop track and note that a fast vehicle passes a stationary observer more often than does a slow one.

2.1.4 Generalized Definitions of Traffic Stream Properties. To describe a traffic stream, one usually seeks to measure properties that are not sensitive to the variations in the individual objects (e.g. the vehicles or their operators) without averaging-out features of interest. This is the trade-off inherent in choosing between short and long measurement intervals, as previously noted. It was partly to address this trade-off that Edie (1965, 1974) proposed some generalized definitions of flow and density that averaged these properties in the manner described below. To begin this discussion, the thin, horizontal rectangle in Figure 2-2 corresponds to a fixed observation point. As per its conventional definition provided earlier, the flow at this point is m / T, where m = 4 in the figure. Since this point in space is a region of temporal duration T and elemental spatial dimension dx, the flow can be expressed equivalently as .

dxTdxm

�� The denominator is the

euclidean area of the thin horizontal rectangle, expressed in units of distance � time. The numerator


2-4

Figure 2-2. Trajectories in time-space region.

is the total distance traveled by all objects in this thin region, since objects cannot enter or exit the region via its elementally small left and right sides.

That flow, then, is the ratio of the distance traveled in a region to the region’s area is valid for any time-space region, since all regions are composed of elementary rectangles. Taking, for example, region A in Figure 2-2, Edie’s generalized definition of the flow in A, q(A), is d(A) / |A|, where d(A) is the total distance traveled in A and |A| is used to denote the region’s area. As the analogue to this, the thin, vertical rectangle in Figure 2-2 corresponds to an instant in time. As per its conventional definition, density is n / L (where n = 2 in this figure) and this can be expressed equivalently as .

dtLdtn

�� It follows that Edie’s generalized definition of density in a region A,

k(A), is t(A) / |A|, where t(A) is the total time spent in A. It should be clear that these generalized definitions merely average the flows collected over all points, and the densities collected at each instant, within the region of interest. Dividing this flow by this density gives d(A) / t(A), which can be taken as the average velocity of objects in A, v(A). The reader will note that, with Edie’s definitions, the average velocity is the ratio of flow to density. Traffic measurement devices, such as loop detectors installed beneath the road surface, can be used to measure flows, densities and average vehicle velocities in ways that are consistent with these generalized definitions. Discussion of this is offered in Cassidy and Coifman (1997). As a final note regarding v(A), when A is taken as a thin horizontal rectangle of spatial dimension dx, the time spent in the region by object i is dx / vi, where vi is i’s velocity. Thus, for this thin,


2-5

horizontal region A, t(A) = .11

��

m

i ivdx Given that for the same region, d(A) = m·dx, the generalized

mean velocity becomes (2.12) ,11

1)()()(

1�

��

�

m

i ivmAtAdAv

i.e., the reciprocal of the mean of the reciprocal velocities, or the harmonic mean velocity. The 1 / vi is often referred to as the pace of i, pi, and thus

.1)(1

1

��

��

��m

iip

mAv

(2.13) Eq. 2.15 applies for regions with L > dx provided that all i span the L and that each pi (or vi) is i’s average over the L. It follows that when conditions in a region A are stationary, the harmonic mean of the velocities measured at a fixed point in A is the v(A). By the same token, the v(A) is the space-mean velocity measured at any instant in A (provided, again, that conditions are stationary).

2.1.5 The Relation Between Density and Occupancy. Occupancy is conventionally defined as the percentage of time that vehicles spend atop a loop detector. It is a commonly-used property for describing highway traffic streams; it is used later in this chapter, for example, for diagnosing freeway traffic conditions. In particular, occupancy is a proxy for density. The following discussion demonstrates that the former is merely a dimensionless version of the latter. One can readily demonstrate this relation by adopting a generalized definition of occupancy analogous to the definitions proposed by Edie. Such a definition is made evident by illustrating each trajectory with two parallel lines tracing the vehicle’s front and rear (as seen by a detector) and this is exemplified in Figure 2-3. The (generalized) occupancy in the region A, �(A), can be taken as the fraction of the region’s area covered by the shaded strips in the figure. From this, it follows that the �(A) and the k(A) are related by an average of the vehicle lengths. This average vehicle length is, by definition, the area of the shaded strips within A divided by the t(A); i.e., it is the ratio of the �(A) to the k(A),

.

)(||

||)()(

AtA

A strips shadedthe of area

AkA length vehicle average ��

�� (2.14)


2-6

Figure 2-3. Trajectories of vehicle fronts and rears.

Notably, an average of the vehicle lengths also relates the k(A) to �, where the latter is the occupancy as conventionally defined (i.e., the percentage of time vehicles spend atop the detector). Toward illustrating this relation, the L in Figure 6-4 is assumed to be the length of road “visible” to the loop detector, the so-called detection zone. The T is some interval of time; e.g. the interval over which the detector collects measurements. The time each ith vehicle spends atop the detector is denoted as �i. Thus, if m vehicles pass the detector during time T, the

.1

T

m

ii��

��

As shown in Figure 6-4, L i is the summed length of the detection zone and the length of vehicle i.Therefore,

� ��

��m

i

m

iii

ii

m

ii p

v1 11

1 L L� (2.15)

if the front end of each i has a constant vi over the distance L i. Since

,1)(1

1

1

11 ��

��

��

�

��m

ii

m

ii

m

ii

mAq

Tm

mT

(2.16)


2-7

it follows that

,)(

,1)()(

1)(

,1)(

1

1

1

1

��

�

�

��

��

��

�

��

��

�

�

�

�

�

�

�

�

m

ii

m

iii

m

iii

m

iii

p

pAk

pm

AvAv

Aq

pm

Aq

L

L

L

�

�

�

(2.17)

where the term in brackets is the average vehicle length relating � to the k(A); it is the so-called average effective vehicle length weighted by the paces. If pace and vehicle length are uncorrelated, the term in brackets in (2.17) can be approximated by the unweighted average of the vehicle lengths in the interval T.

When measurements are taken by two closely spaced detectors, a so-called speed trap, the piare computed from each vehicle’s arrival times at the two detectors. The Li are thus computed by assuming that the pi are constant over the length of the speed trap. When only a single loop detector is available, vehicle velocities are often estimated by using an assumed average value of the (effective) vehicle lengths.

2.1.6 Three-Dimensional Representation of Vehicle Streams. It is useful to display flows and densities using a three-dimensional representation described by Makagami et al. (1971).For this representation, an axis for the cumulative number of objects, N, is added to the t-x coordinate system so that the resulting surface N(t, x) is like a staircase with each trajectory being the edge of a step. As shown in Figure 2-4, curves of cumulative count versus time are obtained by taking cross-sections of this surface at some fixed locations and viewing the exposed regions in the t-N plane. Analogously, cross-sections at fixed times viewed in the N-x plane reveal curves of cumulative count versus space.

Figure 2-4 shows cumulative curves at two locations and for two instants in time. The former display the trip times of objects and the time-varying accumulations between the two locations, as labeled on the figure. These cumulative curves can be transformed into a queueing diagram (as described in Chapter 5) by translating the curve at upstream x1forward by the free-flow (i.e., the undelayed) trip time from x1 to x2. Also displayed in Figure 2-4, the curves of cumulative count versus space show the number of objects crossing a fixed location during the interval t2 - t1 and the distances traveled by individual objects during this same interval.

If one is dealing with many objects so that measuring the exact integer numbers is not important, it is advantageous to construct the cumulative curves with piece-wise linear approximations; e.g. the curves may be smoothed using linear interpolations that pass through the crests of the steps. The time-dependent flows past some location are the slopes of the smoothed curve of t versus N


2-8

constructed at that location (Moskowitz, 1954; Edie and Foote, 1960; Newell, 1971, 1982). Analogously, the location-dependent densities at some instant are the negative slopes of a smoothed curve of N versus x; densities are the negative slopes because objects are numbered in the reverse direction to their motion. This three-dimensional model has been applied by Newell (1993) in work on kinematic waves in traffic (see Chapter 5). In addition, Part I of his paper contains some historical notes on the use of this approach for modelling.

Figure 2-4. Three-dimensional representation.

2.2 Measurement issues This section addresses issues related to the measurement of the key variables defined in the previous section. Four topics are covered. First, there is a discussion of measurement procedures that have traditionally been used. Second, potential measurement errors arising from the mismatch between the definitions and the measurement methods are discussed. Third, measurement difficulties that can potentially arise from the particular location used for collecting measurements are considered. The final topic is the nature of the time intervals from which to collect data, drawing on the cumulative curves just presented. 2.2.1 Measurement procedures Measurement technology for obtaining traffic data has


2-9

changed over the 60-year span of interest in traffic flow, but most of the basic procedures remain largely the same. Five measurement procedures are discussed in this section: � measurement at a point; � measurement over a short section (by which is meant less than about 10 meters (m); � measurement over a length of road [usually at least 0.5 kilometers (km)]; � the use of an observer moving in the traffic stream; and � wide-area samples obtained simultaneously from a number of vehicles, as part of Intelligent

Transportation Systems (ITS). The first two were discussed with reference to the time-space diagram in Fig 2.2, and the remaining three are also readily interpreted on that diagram. Details on each of these methods can be found in the ITE's Manual of Transportation Engineering Studies (Robertson, 1994). The wide-area samples from ITS are similar to having a number of moving observers at various points and times within the system. New developments such as this will undoubtedly change the way some traffic measurements are obtained in the future, but they have not been in operation long enough to have a major effect on the material to be covered in this chapter. Measurement at a point by hand tallies or pneumatic tubes, was the first procedure used for traffic data collection. This method is easily capable of providing volume counts and therefore flow rates directly, and can provide headways if arrival times are recorded. The technology for making measurements at a point on freeways changed over 30 years ago from using pneumatic tubes placed across the roadway to using point detectors (May et al. 1963; Athol 1965). The most commonly used point detectors are based on inductive loop technology, but other methods in use include microwave, radar, photocells, ultrasonics, and closed circuit television cameras. Except for the case of a stopped vehicle, speeds at a 'point' can be obtained only by radar or microwave detectors: dx/dt obviously requires some dx, however small. (Radar and microwave frequencies of operation mean that a vehicle needs to move only about one centimeter during the speed measurement.) In the absence of such instruments for a moving vehicle, a second observation location is necessary to obtain speeds, which moves the discussion to that of measurements over a short section. Measurements over a short section Early studies used a second pneumatic tube, placed very close to the first, to obtain speeds. More recent systems have used paired presence detectors, such as inductive loops spaced perhaps five to six meters apart. With video camera technology, two detector 'lines' placed close together provide the same capability for measuring speeds. All of these presence detectors continue to provide direct measurement of volume and of time headways, as well as of speed when pairs of them are used. Most point detectors currently used, such as inductive loops or microwave beams, take up space on the road, and are therefore a short section measurement. These detectors also measure occupancy, which was not available from earlier technology. This variable is available because the loop gives a continuous reading (at 50 or 60 Hz usually), which pneumatic tubes or manual


2-10

counts could not do. Because occupancy depends on the size of the detection zone of the instrument, the measured occupancy may differ from site to site for identical traffic, depending on the nature and construction of the detector. Measurements along a length of road come either from aerial photography, or from cameras mounted on tall buildings or poles. On the basis of a single frame from such sources, only density can be measured. The single frame gives no sense of time, so neither volumes nor speed can be measured. Once several frames are available, as from a video-camera or from time-lapse photography over short time intervals, speeds and volumes can also be measured, as per the generalized definitions provided above. Despite considerable improvements in technology, and the presence of closed circuit television on many freeways, there is very little use of measurements taken over a long section at the present time. The one advantage such measurements might provide would be to yield true journey times over a lengthy section of road, but that would require better computer vision algorithms (to match vehicles at both ends of the section) than are currently possible. There have been some efforts toward the objective of collecting journey time data on the basis of the details of the 'signature' of particular vehicles or platoons of vehicles across a series of loops over an extended distance (Kühne and Immes 1993), but few practical implementations as yet. The moving observer method was used in some early studies, but is not used as the primary data collection technique now because of the prevalence of the other technologies. There are two approaches to the moving observer method. The first is a simple floating car procedure in which speeds and travel times are recorded as a function of time and location along the road. While the intention in this method is that the floating car behaves as an average vehicle within the traffic stream, the method cannot give precise average speed data. It is, however, effective for obtaining qualitative information about freeway operations without the need for elaborate equipment or procedures. One form of this approach uses a second person in the car to record speeds and travel times. A second form uses a modified recording speedometer of the type regularly used in long-distance trucks or buses. One drawback of this approach is that it means there are usually significantly fewer speed observations than volume observations. An example of this kind of problem appears in Morton and Jackson (1992). The other approach was developed by Wardrop and Charlesworth (1954) for urban traffic measurements and is meant to obtain both speed and volume measurements simultaneously. Although the method is not practical for major urban freeways, it is included here because it may be of some value for rural expressway data collection, where there are no automatic systems. While it is not appropriate as the primary mode of data collection on a busy freeway, there are some useful points that come out of the literature that should be noted by those seeking to obtain average speeds through the moving car method. The method developed by Wardrop and Charlesworth is based on a survey vehicle that travels in both directions on the road. The formulae allow one to estimate both speeds and flows for one direction of travel. The two formulae are


2-11

) t + t(y) + (x = qwa

(2.18)

qytt w � (2.19)

where q is the estimated flow on the road in the direction of interest,

x is the number of vehicles traveling in the direction of interest, which are met by the survey vehicle while traveling in the opposite direction,

y is the net number of vehicles that overtake the survey vehicle while traveling in the direction of interest (i.e. those passing minus those overtaken),

ta is the travel time taken for the trip against the stream, tw is the travel time for the trip with the stream, and

t is the estimate of mean travel time in the direction of interest. Wright (1973) revisited the theory behind this method. His paper also serves as a review of the papers dealing with the method in the two decades between the original work and his own. He finds that, in general, the method gives biased results, although the degree of bias is not significant in practice, and can be overcome. Wright's proposal is that the driver should fix the journey time in advance, and keep to it. Stops along the way would not matter, so long as the total time taken is as determined prior to travel. Wright's other point is that turning traffic (exiting or entering) can upset the calculations done using this method. This fact means that the route to be used for this method needs to avoid major exits or entrances. It should be noted also that a large number of observations are required for reliable estimation of speeds and flows; without that, the method has very limited precision. 2.2.2 Error caused by the mismatch between definitions and usual measurements The overview in the previous section described methods that have historically been used to collect observations on the key traffic variables. As was mentioned within that section, the methods do not always accord with the definitions of these variables, as they were presented in Section 2.1. One of the most common methods for collecting these data currently is based on inductive loops embedded in the roadway. When speed data are also to be collected, a pair of closely spaced loops (often called a speed trap) is needed, a known distance apart. Equivalent systems are based on microwave beams that cover a part of a roadway surface. Cassidy and Coifman (1997) point out the criteria that need to be met for the data from these systems to meet the definitional requirements.

In practical terms, these criteria can be reduced to ensuring that any vehicle entering the speed trap also clears it within the time interval for which the data are obtained – or that the number of vehicles in the time interval is quite large. Neither of these criteria is met by the ordinary implementation of inductive loop speed trap detectors. Whether or not the last vehicle


2-12

entering a speed trap will clear it within the interval is a simple random variable. There is no guarantee that this criterion will regularly be met. If the number of vehicles were very large, this would not be an issue, but because of the need for timely information, many systems poll the loop detector controllers at least every minute, with some systems going to 30 second or even 20 second data collection. For intervals of that size, the volume counts on a single freeway lane will be at most 40 (or 25 or 20) vehicles, which is not large enough to overcome the error introduced by missing one of the vehicle’s speeds. While the data from these detector systems are certainly good enough for operational decision-making, the data may give misleading results if used directly with these equations because of the mismatch between the collection and the definitions. 2.2.3 Importance of location to the nature of the data Almost all of the bivariate models to be discussed represent efforts to explain the behaviour of traffic variables over the full range of operation. In turning the models from abstract representations into numerical models with specific parameter values, an important practical question arises: can one expect that the data collected will cover the full range that the model is intended to cover? If the answer is no, then the difficult question follows of how to do curve fitting (or parameter estimation) when there may be essential data missing. This issue can be explained more easily with an example. At the risk of oversimplifying a relationship prior to a more detailed discussion of it, consider the simple representation of the speed-flow curve as shown in Figure 2.5, for three distinct sections of roadway. The underlying curve is assumed to be the same at all three locations. Between locations A and B, a major entrance ramp adds considerable traffic to the road. If location B reaches capacity due to this entrance ramp volume, there will be a queue of traffic on the mainstream at location A. These vehicles can be considered to be waiting their turn to be served by the bottleneck section immediately downstream of the entrance ramp. The data superimposed on graph A reflect the situation whereby traffic at A had not reached capacity before the added ramp flow caused the backup. There is a good range of uncongested data (on the top part of the curve), and congested data concentrated in one area of the lower part of the curve. The flows for that portion reflect the capacity flow at B less the entering ramp flows. At location B, the full range of uncongested flows is observed, right out to capacity, but the location never becomes congested, in the sense of experiencing stop-and-go traffic. It does, however, experience congestion in the sense that speeds are below those observed in the absence of the upstream congestion. Drivers arrive at the front end of the queue moving very slowly, and accelerate away from that point, increasing speed as they move through the bottleneck section. This segment of the speed-flow curve has been referred to as queue discharge flow, QDF (Hall et al. 1992). The particular speed observed at B will depend on how far it is from the front end of the queue (Persaud and Hurdle 1988). Consequently, the only data that will be observed at B are on the top portion of the curve, and at some particular speed in the QDF segment. If the exit ramp between B and C removes a significant portion of the traffic that was observed at B, flows at C will not reach the levels they did at B. If there is no downstream situation similar to that between A and B, then C will not experience congested operations, and


2-13

the data observable there will be as shown in Figure 2.5. None of these locations taken alone can provide the data to identify the full speed-flow curve. Location C can help to identify the uncongested portion, but cannot deal with capacity, or with congestion. Location B can provide information on the uncongested portion and on capacity operation, but cannot contribute to the discussion of congested operations. Location A can provide some information on both uncongested and congested operations, but cannot tell anything about capacity operations. This would all seem obvious enough. A similar discussion appears in Drake et al (1967). It is also explained by May (1990). Other aspects of the effect of location on data patterns are discussed by Hsu and Banks (1993). Yet a number of important efforts to fit data to theory have ignored this key point (for example Ceder and May 1976; Easa and May 1980). They have recognized that location-A data are needed to fit the congested portion of the curve, but have not recognized that at such a location data are missing that are needed to identify capacity. Consequently, discussion of bivariate models will look at the nature of the data used in each study, and at where the data were collected (with respect to bottlenecks) in order to evaluate the theoretical ideas. It is possible that the apparent need for several different models, or for different parameters for the same model at different locations, or even for discontinuous models instead of continuous ones, arose because of the nature (location) of the data each was using. 2.2.4 Selecting intervals from which to extract data

In addition to the location for the data, there are also several issues relating to the time intervals for collecting data. The first is the issue of obtaining representative data. By examining trends on the cumulative curves, one can observe how flows and densities change with time and space. By selecting flows and densities as they appear on the cumulative curves, their values may be taken over intervals that exhibit near-constant slopes. In this way, the values assigned to these properties are not affected by some arbitrarily selected measurement interval(s). Choosing intervals arbitrarily is undesirable because data extracted over short measurement intervals are highly susceptible to the effects of statistical fluctuations while the use of longer intervals may average-out the features of interest. Further discussion and demonstration of this in the context of freeway traffic is offered in (Cassidy, 1998). There is also an issue of how many observations are needed to obtain good estimates of key variables such as the capacity. A bottleneck’s capacity, qmax , is the maximum flow it can sustain for a very long time (in the absence of any influences from restrictions further downstream). It can be expressed mathematically as

,lim max

max ��

��

��

�� TN

qT (2.20)

where Nmax denotes that the vehicles counted during very long time T discharged through the bottleneck at a maximum rate. The engineer assigns a capacity to a bottleneck by obtaining a value for the estimator qmax (since one cannot actually observe a maximum flow for a time period approaching infinity). It is desirable that the expected value of this estimator equal the capacity, E(qmax) � qmax. For this reason, one would collect samples (i.e., counts) immediately downstream of an active bottleneck so as to measure vehicles discharging at a maximum rate. The amount qmax can


2-14

deviate from qmax is controlled by the sample size, N. A formula for determining N to estimate a bottleneck’s capacity to a specified precision is derived below.

To begin this derivation, the estimator may be taken as

(2.21) where nm is the count collected in the mth interval and each of these M intervals has a duration of �. If the {nm} can be taken as independent, identically distributed random variables (e.g. the counts were collected from consecutive intervals with � sufficiently large), then the variance of qmax can be expressed as

^

^

��

��

��

� ��

��

)variance(nTM

)variance(n)(qmax^variance 11

2

� ��

M

mm Mnmaxq ,)/(

1

(2.22)

since qmax is a linear function of the independent nm and the (finite) observation period T is the denominator in (2.21).

The bracketed term variance(n) / � is a constant. Thus, by multiplying the top and bottom of this quotient by E(n), the expected value of the counts, and by noting that E(n) / � = qmax , one obtains

(2.23)

where � is the index of dispersion; i.e., the ratio variance(n)/E(n). The variance(qmax ) is the square of the standard error. Thus, by isolating T in (2.23) and then

multiplying both sides of the resulting expression by qmax , one arrives at

(2.24)

where qmax T � N, the number of observations (i.e., the count) needed to estimate capacity to a specified percent error �. Note, for example, that � = 0.05 to obtain an estimate within 5 percent of qmax. The value of � may be estimated by collecting a presample and, notably, N increases rapidly as � diminishes.

^

The expression N = �/�2 may be used to determine an adequate sample size when vehicles, or any objects, discharging through an active bottleneck exhibit a nearly stationary flow; i.e., when the cumulative count curve exhibits a nearly constant slope. If necessary, the N samples may be obtained by concatenating observations from multiple days. Naturally, one would take samples during time periods thought to be representative of the conditions of interest. For example, one should probably not use vehicle counts taken in inclement weather to estimate the capacity for fair weather conditions. 2.3 Bivariate models

This section provides an overview of work to establish relationships among pairs of the variables described in the opening section. Some of these efforts begin with mathematical models; others are primarily empirical, with little or no attempt to generalize. Both are important components for understanding the relationships. For reasons discussed above, two aspects of these efforts are emphasized here: the measurement methods used to obtain the data; and the location at which the measurements were obtained.

^

,2max ��

��Tq

,)(variance max T�

�q^


2-15

2.3.1 Speed-flow models The speed-flow relationship is the bivariate relationship on which there has been the greatest amount of work recently, so is the first one discussed. This section starts from current understanding, which provides some useful insights for interpreting earlier work, and then moves to a chronological review of some of the major models. The bulk of then recent empirical work on the relationship between speed and flow (as well as the other relationships) was summarized in a paper by Hall, Hurdle, and Banks (1992). In it, they proposed the model for traffic flow shown in Figure 2.6. This figure is the basis for the background speed-flow curve in Figure 2.5. It is useful to summarize some of the antecedents of the understanding depicted in Figure 2.6. The initial impetus came from a paper by Persaud and Hurdle (1988), in which they demonstrated (Figure 2.7) that the vertical line for queue discharge flow in Figure 2.6 was a reasonable result of measurements taken at various distances downstream from a queue. This study was an outgrowth of an earlier one by Hurdle and Datta (1983) in which they raised a number of questions about the shape of the speed-flow curve near capacity. Additional empirical work dealing with the speed-flow relationship was conducted by Banks (1989, 1990), Hall and Hall (1990), Chin and May (1991), Wemple, Morris and May (1991), Agyemang-Duah and Hall (1991) and Ringert and Urbanik (1993). All of these studies supported the idea that speeds remain nearly constant even at quite high flow rates. Another of the important issues they dealt with is one that had been around for over thirty years (Wattleworth 1963): is there a reduction in flow rates within the bottleneck at the time that the queue forms upstream? Figure 2.6 shows such a drop on the basis of two studies. Banks (1991a, 1991b) reports roughly a 3% drop from pre-queue flows, on the basis of nine days of data at one site in California. Agyemang-Duah and Hall (1991) found about a 5% decrease, on the basis of 52 days of data at one site in Ontario. This decrease in flow is often not observable, however, as in many locations high flow rates do not last long enough prior to the onset of congestion to yield the stable flow values that would show the drop. Two empirical studies in Germany support the upper part of the curve in Figure 2.6 quite well. Heidemann and Hotop (1990) found a piecewise-linear 'polygon' for the upper part of the curve (Figure 2.8). Unfortunately, they did not have data beyond 1700 veh/hr/lane, so could not address what happens at capacity. Stappert and Theis (1990) conducted a major empirical study of speed-flow relationships on various kinds of roads. However, they were interested only in estimating parameters for a specific functional form,

V = (A - eBQ) e-c - K edQ (2.25)

where V is speed, Q is traffic volume, c and d are constant "Krummungs factors" taking values between 0.2 and 0.003, and A, B, and K are parameters of the model. This function tended to


2-16

�

�

�


2-17

�

�


2-18

give the kind of result shown in Figure 2.9, despite the fact that the curve does not accord well with the data near capacity. In Figure 2.9, each point represents a full hour of data, and the graph represents five months of hourly data. Note that flows in excess of 2200 veh/h/lane were sustained on several occasions, over the full hour. The problem for traffic flow theory is that these curves are empirically derived. There is not really any theory that would explain these particular shapes, except perhaps for Edie et al.(1980), who propose qualitative flow regimes that relate well to these curves. The task that lies ahead for traffic flow theorists is to develop a consistent set of equations that can replicate this reality. The models that have been proposed to date, and will be discussed in subsequent sections, do not necessarily lead to the kinds of speed-flow curves that data suggest are needed. It is instructive to review the history of depictions of speed-flow curves in light of this current understanding. Probably the seminal work on this topic was the paper by Greenshields in 1935, in which he derived the following parabolic speed-flow curve on the basis of a linear speed-density relationship together with the equation, flow = speed * density:

q = kj (u -u2uf

) (2.26)

Figure 2.10 contains that curve, and the data it is based on, redrawn. The numbers against the data points represent the "number of 100-vehicle groups observed", on Labor Day 1934, in one direction on a two-lane two-way road (p. 464). In counting the vehicles on the road, every 10th vehicle started a new group (of 100), so there is 90% overlap between two adjacent groups (p. 451): the groups are not independent observations. Equally important, the data have been grouped in flow ranges of 200 veh/h and the averages of these groups taken prior to plotting. The one congested point, representing 51 (overlapping) groups of 100 observations, came from a different roadway entirely, with different cross-section and pavement, which were collected on a different day. These details are mentioned here because of the importance to traffic flow theory of Greenshields' work. The parabolic shape he derived was accepted as the proper shape of the curve for decades. In the 1965 Highway Capacity Manual, for example, the shape shown in Figure 2.10 appears exactly, despite the fact that data displayed elsewhere in the 1965 HCM showed that contemporary empirical results did not match the figure. In the 1985 HCM, the same parabolic shape was retained, although broadened considerably. In short, Greenshields' model dominated the field for over 50 years, despite the fact that by current standards of research the method of analysis of the data, with overlapping groups and averaging prior to curve-fitting, would not be acceptable. There is another criticism that can be addressed to Greenshields' work as well, although it is one of which a number of current researchers seem unaware. Duncan (1976; 1979) has shown that calculating density from speed and flow, fitting a line to the speed-density data, and then converting that line into a speed-flow function, gives a biased result relative to direct estimation


2-19

�

�


2-20

of the speed-flow function. This is a consequence of three things discussed earlier: the non-linear transformations involved in both directions, the stochastic nature of the observations, and the inability to match the time and space measurement frames exactly. It is interesting to contrast the emphasis on speed-flow models in recent years with that 20 years ago, as represented in TRB SR 165 (Gerlough and Huber 1975). In that volume, the discussion of speed-flow models took up less than a page of text, and none of the five accompanying diagrams dealt with freeways. (Three dealt with the artificial situation of a test track.) In contrast, five pages and eleven figures were devoted to the speed-density relationship. Speed-flow models are important for freeway management strategies, and will be of fundamental importance for intelligent vehicle/highway systems (IVHS) implementation of alternate routing; hence there is considerably more work on this topic than on the remaining two bivariate topics. Twenty and more years ago, the other topics were of more interest. As Gerlough and Huber stated the matter (p. 61), "once a speed-concentration model has been determined, a speed-flow model can be determined from it." That is in fact the way most earlier speed-flow work was treated (including that of Greenshields). Hence it is sensible to turn to discussion of speed-concentration models, and to deal with any other speed-flow models as a consequence of speed-concentration work, which is the way they were developed. 2.3.2 Speed-concentration models Greenshields' (1935) linear model of speed and density was mentioned in the previous section. It is:

u = uf (1-k /kj) (2.27)

where uf is the free-flow speed, and kj is the jam density. The measured data were speeds and flows; density was calculated using equation 2.27. The most interesting aspect of this particular model is that its empirical basis consisted of half a dozen points in one cluster near free-flow speed, and a single observation under congested conditions (Figure 2.11). The linear relationship comes from connecting the cluster with the single point: "since the curve is a straight line it is only necessary to determine accurately two points to fix its direction" (p. 468). What is surprising is not that such simple analytical methods were used in 1935, but that their results (the linear speed-density model) have continued to be so widely accepted for so long. While there have been studies that claimed to have confirmed this model, such as that in Figure 2.12a (Huber 1957), they tended to have similarly sparse portions of the full range of data, usually omitting both the lowest flows, and flow in the range near capacity. There have also been a number of studies that found contradictory evidence, most importantly that by Drake, et al. (1965), which will be discussed in more detail subsequently. A second early model was that put forward by Greenberg (1959), showing a logarithmic relationship:


2-21

u = umln(k/kj) (2.28)

The paper showed the fit of the model to two data sets, both of which visually looked very reasonable. However, the first data set was derived from speed and headway data on individual vehicles, which "was then separated into speed classes and the average headway was calculated for each speed class" (p. 83). In other words, the vehicles that appear in one data point (speed class) may not even have been travelling together! While a density can always be calculated as the reciprocal of average headway, when that average is taken over vehicles that may well not have been travelling together, it is not clear what that density is meant to represent. The second data set used by Greenberg was Huber's. This is the same data that appears in Figure 2.12a; Greenberg's graph is shown in Figure 2.12b. Visually, the fit is quite good, but Huber reported an R of 0.97, which does not leave much room for improvement. A third model from the same period is that by Edie (1961). His model was an attempt to deal with a discontinuity that had consistently been found in data near the critical density, which “suggested the existence of some kind of change of state” (p. 72). He proposed two linear models for the two states of flow. The first related density to the logarithm of velocity above the “optimum velocity”, i.e. “non-congested flow”. The second related velocity to the logarithm of spacing (i.e. the inverse of density) for congested flow. The model was fitted to the same Lincoln Tunnel data as used by Greenberg. These three forms of the speed-density curve, plus four others, were investigated in an empirical test by Drake et al. in 1967. The test used data from the middle lane of the Eisenhower Expressway in Chicago, one-half mile (800 m) upstream from a bottleneck whose capacity was only slightly less than the capacity of the study site. This location was chosen specifically in order to obtain data over as much of the range of operations as possible. A series of 1224 1-minute observations were initially collected. The measured data consisted of volume, time-mean speed, and occupancy. Density was calculated from volume and time-mean speed. A sample was then taken from among the 1224 data points in order to create a data set that was uniformly distributed along the density axis, as is assumed by regression analysis of speed on density. The intention in conducting the study was to compare the seven speed-density hypotheses statistically, and thereby to select the best one. In addition to Greenshields' linear form, Greenberg's exponential curve, and Edie’s two-part logarithmic model, the other four investigated were a two-part and three-part piecewise linear models, Underwood's (1961) transposed exponential curve, and a bell-shaped curve. Despite the intention to use "a rigorous structure of falsifiable tests" (p. 75) in this comparison, and the careful way the work was done, the statistical analyses proved inconclusive: "almost all conclusions were based on intuition alone since the statistical tests provided little decision power after all" (p. 76). To assert that intuition alone was the basis is no doubt a bit of an exaggeration. Twenty-one graphs help considerably in differentiating among the seven hypotheses and their consequences for both speed-volume and volume-density graphs.


2-22

�


2-23

Figure 2.13 provides an example of the three types of graphs used, in this case the ones based on the Edie model. Their comments about this model (p. 75) were: "The Edie formulation gave the best estimates of the fundamental parameters. While its R2 was the second lowest, its standard error was the lowest of all hypotheses." One interesting point with respect to Figure 2.13 is that the Edie model was the only one of the seven to replicate capacity operations closely on the volume-density and speed-volume plots. The other models tended to underestimate the maximum flows, often by a considerable margin, as is illustrated in Figure 2.14, which shows the speed-volume curve resulting from Greenshields' hypothesis of a linear speed-density relationship. (It is interesting to note that the data in these two figures are quite consistent with the currently accepted speed-flow shape identified earlier in Figures 2.5 and 2.6.) The overall conclusion one might draw from the Drake et al. study is that none of the seven models they tested provide a particularly good fit to or explanation of the data, although it should be noted that they did not state their conclusion this way, but rather dealt with each model separately. There are four additional issues that arise from the Drake et al. study that are worth noting here. The first is the methodological one identified by Duncan (1976; 1979), and discussed earlier with regard to Greenshields' work. Duncan showed that calculating density from speed and flow data, fitting a speed-density function to that data, and then transforming the speed-density function into a speed-flow function results in a curve that does not fit the original speed-flow data particularly well. This is the method used by Drake et al, and certainly most of their resulting speed-flow functions did not fit the original speed-flow data very well. Duncan's 1979 paper expanded on the difficulties to show that minor changes in the speed-density function led to major changes in the speed-flow function, suggesting the need for further caution in using this method of double transformations to get a speed-flow curve.

The second is that of the data collection location, as discussed above. The data were collected upstream of a bottleneck, which produced the kind of discontinuity that Edie had earlier identified. The Drake et al. approach was to try to fit the data as they had been obtained, without considering whether there was a portion of the data that was missing. They had intentionally tried to obtain data from as wide a range as possible, but as discussed above it may not be possible to get data from all three parts of the curve at one location.

The third issue is that identified by Cassidy, relating to the time period for collection of

the data. The Drake et al. data came from standard loop detectors, working on fixed time intervals. As a consequence, there will be some measurement error, which may well affect the estimation of the bivariate relationships. The fourth issue is the relationship between car-following models (see Chapter 5) and the models tested by Drake et al. They noted that four of the models they tested "have been shown to be directly related to specific car-following rules", and cited articles by Gazis and co-authors (1959; 1961). The interesting question to raise in the context of the overall appraisal of the Drake et al. results is whether those results raise doubts about the validity of the car-following models for freeways. The car-following models gave rise to four of the speed-density models tested by Drake et al. The results of the testing suggest that the speed-density models are not


2-24

�


2-25

particularly good. Modus tollens in logic says that if the consequences of a set of premises are shown to be false, then one (at least) of the premises is not valid. It is possible, then, that the car-following models are not valid for freeways. (This is not surprising, as they were not developed for this context. Nor, it seems, are they used as the basis for contemporary microscopic freeway simulation models.) On the other hand, any of the three issues just identified may be the source of the failure of the models, rather than their development from car-following models.

2.3.3 Flow-concentration models Although Gerlough and Huber did not give the topic of flow-concentration models such extensive treatment as they gave the speed-concentration models, they nonetheless thought this topic to be very important, as evidenced by their introductory paragraph for the section dealing with these models (p. 55): Early studies of highway capacity followed two principal approaches. Some

investigators examined speed-flow relationships at low concentrations; others discussed headway phenomena at high concentrations. Lighthill and Whitham [1955] have proposed use of the flow-concentration curve as a means of unifying these two approaches. Because of this unifying feature, and because of the great usefulness of the flow-concentration curve in traffic control situations (such as metering a freeway), Haight [1960; 1963] has termed the flow-concentration curve "the basic diagram of traffic".

Nevertheless, most flow-concentration models have been derived from assumptions about the shape of the speed-concentration curve. This section deals primarily with work that has focused on the flow-concentration relationship directly. Under that heading is included work that uses either density or occupancy as the measure of concentration. Edie was perhaps the first to point out that empirical flow-concentration data frequently have discontinuities in the vicinity of what would be maximum flow, and to suggest that therefore discontinuous curves might be needed for this relationship. (An example of his type of curve appears in Figure 2.13 above.) This suggestion led to a series of investigations by May and his students (Ceder 1975; 1976; Ceder and May 1976; Easa and May 1980) to specify more tightly the nature and parameters of these "two-regime" models (and to link those parameters to the parameters of car-following models). The difficulty with their resulting models is that the models often do not fit the data well at capacity (with results similar to those shown in Figure 2.14 for Greenshields' single-regime model). In addition, there seems little consistency in parameters from one location to another. Even more troubling, when multiple days from the same site were calibrated, the different days required quite different parameters. Koshi, Iwasaki and Ohkura (1983) gave an empirically-based discussion of the flow-density relationship, in which they suggested that a reverse lambda shape was the best description of the data (p. 406): "the two regions of flow form not a single downward concave


2-26

curve... but a shape like a mirror image of the Greek letter lamda [sic] (�)". These authors also investigated the implications of this phenomenon for car-following models, as well as for wave propagation. Data with a similar shape to theirs appears in Figure 2.13; Edie’s equations fit those data with a shape similar to the lambda shape Koshi et al. suggested. Although most of the flow-concentration work that relies on occupancy rather than density dates from the past decade, Athol suggested its use nearly 30 years earlier (in 1965). His work presages a number of the points that have come out subsequently and are discussed in more detail below: the use of volume and occupancy together to identify the onset of congestion; the transitions between uncongested and congested operations at volumes lower than capacity; and the use of time-traced plots (i.e. those in which lines connected the data points that occurred consecutively over time) to better understand the operations. After Athol's early efforts, there seems to have been a dearth of efforts to utilize the occupancy data that was available, until the mid-1980s. One paper from that time (Hall et al. 1986) that utilized occupancy drew on the same approach Athol had used, namely the presentation of time-traced plots. Figure 2.15 shows results for four different days from the same location, 4 km upstream of a primary bottleneck. The data are for the left-most lane only (the high-speed, or passing lane), and are for 5-minute intervals. The first point in the time-connected traces is the one that occurred in the 5-minute period after the data-recording system was turned on in the morning. In part D of the Figure, it is clear that operations had already broken down prior to data being recorded. Part C is perhaps the most intriguing: operations move into higher occupancies (congestion) at flows clearly below maximum flows. Although Parts A and B may be taken to confirm the implicit assumption many traffic engineers have that operations pass through capacity prior to breakdown, Part C gives a clear indication that this does not always happen. Even more important, all four parts of Figure 2.15 show that operations do not go through capacity in returning from congested to uncongested conditions. Operations can 'jump' from one branch of the curve to the other, without staying on the curve. (This same result, not surprisingly, occurs for speed-flow data.) Each of the four parts of Figure 2.15 show at least one data point between the two 'branches' of the usual curve during the return to uncongested conditions. Because these were 5-minute data, the authors recognized that these points might be the result of averaging of data from the two separate branches. Subsequently, however, additional work utilizing 30-second intervals confirmed the presence of these same types of data (Persaud and Hall 1989). Hence there appears to be strong evidence that traffic operations on a freeway can move from one branch of the curve to the other without going all the way around the capacity point. This is an aspect of traffic behaviour that none of the mathematical models discussed above either explain or lead one to expect. Nonetheless, the phenomenon has been at least implicitly recognized since Lighthill and Whitham's (1955) discussion of shock waves in traffic, which assumes instantaneous jumps from one branch to the other on a speed-flow or flow-occupancy curve. As well, queueing models (e.g. Newell 1982) imply that immediately upsteam from the back end of a queue there must be points where the speed is changing rapidly from the uncongested branch of the speed-flow curve to that of the congested branch. It would be beneficial if flow-


2-27

�


2-28

concentration (and speed-flow) models explicitly took this possibility into account. One of the conclusions of the paper by Hall et al. (1986) from which Figure 2.15 is drawn is that an inverted 'V' shape is a plausible representation of the flow-occupancy relationship. Although that conclusion was based on limited data from near Toronto, Hall and Gunter (1986) supported it with data from a larger number of stations. Banks (1989) tested their proposition using data from the San Diego area, and confirmed the suggestion of the inverted 'V'. He also offered a mathematical statement of this proposition and a behavioural interpretation of it (p. 58): The inverted-V model implies that drivers maintain a roughly constant average

time gap between their front bumper and the back bumper of the vehicle in front of them, provided their speed is less than some critical value. Once their speed reaches this critical value (which is as fast as they want to go), they cease to be sensitive to vehicle spacing....

2.4 Three-dimensional models There has not been a lot of work that attempts to treat all three traffic flow variables simultaneously. Gerlough and Huber presented one figure (reproduced as Figure 2.16) that represented all three variables, but said little about this, other than (1) "The model must be on the three-dimensional surface u = q/k," and (2) "It is usually more convenient to show the model of [Figure 2.16] as one or more of the three separate relationships in two dimensions..." (p. 49). As was noted earlier, however, empirical observations rarely accord exactly with the relationship q=u k, especially when the observations are taken during congested conditions. Hence focusing on the two-dimensional relationships will not often provide even implicitly a valid three-dimensional relationship. In this context, a paper by Gilchrist and Hall (1989) is interesting because it presents three-dimensional representations of empirical observations, without attempting to fit them to a theoretical representations. The study is limited, in that data from only one location, upstream of a bottleneck, was presented. To enable better visualization of the data, a time-connected trace was used. The projections onto the standard two-dimensional surfaces of the data look much as one might expect. The surprises came in looking at oblique views of the three-dimensional representation, as in Figures 2.17 and 2.18. From one perspective (Figure 2.17), the traditional sideways U-shape that we have been led to expect is quite apparent, and projections of that are easily visualized onto, for example, the speed-flow surface (the face labeled with a 3, on the left side of the ‘box’). From a different perspective (Figure 2.18) that shape is hardly apparent at all, although indications of an inverted 'V' can be seen, which would project onto a flow-occupancy plot on face 1 of the box.. Black and white were alternated for five speed ranges. In both figures, the dark lines at the left represent the data with speeds above 80 km/hr. The light lines closest to the left cover the range 71 to 80 km/hr; the next, very small, area of dark lines contains the range 61 to 70 km/hr; the remaining light lines represent the range 51 to 60 km/hr; and the dark lines to the right of the diagram represent speeds below 50 km/hr.


2-29

�

�


2-30

One recent approach to modelling the three traffic operations variables directly has been based on the mathematics of catastrophe theory. (The name comes from the fact that while most of the variables being modelled change in a continuous fashion, at least one of the variables can make sudden discontinuous changes, referred to as catastrophes by Thom (1975), who originally developed the mathematics for seven such models, ranging from two dimensions to eight.) The first effort to apply these models to traffic data was that by Dendrinos (1978), in which he suggested that the two-dimensional catastrophe model could represent the speed-flow curve. A more fruitful model was proposed by Navin (1986), who suggested that the three-dimensional 'cusp' catastrophe model was appropriate for the three traffic variables. The feature of the cusp catastrophe surface that makes it of interest in the traffic flow context is that while two of the variables (the control variables) exhibit smooth continuous change, the third one (the state variable) can undergo a sudden 'catastrophic' jump in its value. Navin suggested that speed was the variable that underwent this catastrophic change, while flow and occupancy were the control variables. While Navin's presentation was primarily an intuitive one, without recourse to data, Hall and co-authors picked up on the idea and attempted to flesh it out both mathematically and empirically. The initial effort appears in Hall (1987), in which the basic idea was presented, and some data applied. Figure 2.19 is a representation, showing the partially-folded (and torn) surface that is the Maxwell version of the cusp catastrophe surface, approximately located traffic data on that surface, and axes external to the surface representing the general correspondence with traffic variables. Further elaboration of this model is provided by Persaud and Hall (1989). Acha-Daza and Hall (1993) compared the ability of this model with the ability of some of the earlier models discussed above, to estimate speeds from flows obtained using inductive loop detectors and 30-second data, and found the catastrophe theory model to be slightly better than they were. Despite its empirical success, the problem with the model is that it appears to be inconsistent with the basic definitions and relationships with which this chapter opened. 2.5 Summary and links to other chapters The current status of mathematical models for speed-flow-concentration relationships is in a state of flux. The models that dominated the discourse for nearly 30 years are incompatible with the data currently being obtained, and with currently accepted depictions of speed-flow curves, but no replacement models have yet been developed. The analyses of Cassidy and Newell have shown that the data used to develop most of the earlier models were flawed, as has been described above. An additional difficulty was noted by Duncan (1976; 1979): transforming variables, fitting equations, and then transforming the equations back to the original variables can lead to biased results, and is very sensitive to small changes in the initial curve-fitting. Progress has been made in understanding the relationships among the key traffic variables, but there is considerable scope for better models still. It is important to note that the variables and analyses that have been discussed in this chapter are closely related to topics in several of the succeeding chapters. The human factors discussed in Chapter 3, for example, help to explain some of the variability in the data that have


2-31

�

�

Figure 2.19 Catastrophe Theory Surface Showing Sketch of a Possible Freeway Function, and

Projections and Transformations from That (Hall 1987).


2-32

been discussed here. As mentioned earlier, some of the bivariate models discussed above have been derived from car-following models, which are covered in Chapter 4. In addition, brief mention was made here of Lighthill and Witham’s work, which has spawned a large literature related to the behavior of shocks and waves in traffic flow, covered in Chapter 5. All of these topics are inter-related, but have been addressed separately for ease of understanding. Acknowledgements The assistance of Michael Cassidy in developing this latest revision of this chapter was very helpful, as noted in the preface to the chapter. Valuable comments on an early draft of this material were received from Jim Banks, Frank Montgomery and Van Hurdle. The assistance of Richard Cunard and the TRB in bringing this project to completion is also appreciated. References Acha-Daza, J.A. and Hall, F.L. 1993. A graphical comparison of the predictions for speed given by catastrophe theory and some classic models, Transportation Research Record 1398, 119-124. Agyemang-Duah, K. and Hall, F.L. 1991. Some issues regarding the numerical value of freeway capacity, in Highway Capacity and Level of Service, Proceedings of the International Symposium on Highway Capacity, Karlsruhe. U. (Brannolte, ed.) Rotterdam, Balkema, 1-15. Athol, P. 1965. Interdependence of certain operational characteristics within a moving traffic stream. Highway Research Record 72, 58-87. Banks, J.H. 1989. Freeway speed-flow-concentration relationships: more evidence and interpretations, Transportation Research Record 1225, 53-60. Banks, J.H. 1990. Flow processes at a freeway bottleneck. Transportation Research Record 1287, 20-28. Banks, J.H. 1991a. Two-capacity phenomenon at freeway bottlenecks: A basis for ramp metering?, Transportation Research Record 1320, 83-90. Banks, J.H. 1991b. The two-capacity phenomenon: some theoretical issues, Transportation Research Record 1320, pp. 234-241. Cassidy, MJ 1998 Bivariate relations in nearly stationary highway traffic. Transp Res B, 32B, 49-59. Cassidy, MJ 1999 Traffic flow and capacity, in Handbook of transportation science, RW Hall (ed). Norwell, MA: Kluwer Academic Publishers.


2-33

Cassidy , MJ and Coifman, B 1997 Relation among average speed, flow and density and the analogous relation between density and occupancy. Transportation Research Record 1591, 1-6. Ceder, A. 1975. Investigation of two-regime traffic flow models at the micro- and macro-scopic levels. Ph.D. dissertation, University of California, Berkeley, California. Ceder, A. 1976. A deterministic traffic flow model for the two-regime approach. Transportation Research Record 567, 16-32. Ceder, A. and May, A.D. 1976. Further evaluation of single- and two-regime traffic flow models. Transportation Research Record 567, 1-15. Chin, H.C. and May, A.D. 1991. Examination of the speed-flow relationship at the Caldecott Tunnel, Transportation Research Record 1320. Daganzo, CF 1997 Fundamentals of transportation and traffic operations. New York: Elsevier. Dendrinos, D.S. 1978. Operating speeds and volume to capacity ratios: the observed relationship and the fold catastrophe, Transportation Research Record 12, 191-194. Drake, J.S., Schofer, J.L., and May, A.D. 1967. A Statistical Analysis of Speed Density Hypotheses, Highway Research Record 154, 53-87. Duncan, N.C. 1976. A note on speed/flow/concentration relations, Traffic Engineering and Control, 34-35. Duncan, N.C. 1979. A further look at speed/flow/concentration, Traffic Engineering and Control, 482-483. Easa, S.M. and May, A.D. 1980. Generalized procedures for estimating single- and two-regime traffic flow models, Transportation Research Record 772, 24-37. Edie, L.C. 1961. Car following and steady-state theory for non-congested traffic, Operations Research 9, 66-76. Edie, LC 1965 Discussion of traffic stream measurements and definitions. Proc Int Symp. On the Theory of Traffic Flow J. Almond (ed) Paris: OECD, 139-154. Edie, LC 1974 Flow theories, in Traffic Science DC Gazis (ed) New York: Wiley, 8-20. Edie, LC and Foote, R 1960 Effect of shock waves on tunnel traffic flow. Proc Highway Res Board 39, 492-505.


2-34

Edie, L.C., Herman, R., and Lam, T., 1980. Observed multilane speed distributions and the kinetic theory of vehicular traffic, Transportation Science 14, 55-76. Gazis, D.C., Herman, R., and Potts, R. 1959. Car-following theory of steady-state traffic flow. Operations Research 7, 499-505. Gazis, D.C., Herman, R., and Rothery, R.W. 1961. Nonlinear follow-the-leader models of traffic flow. Operations Research 9, 545-567. Gerlough, D.L. and Huber, M.J. 1975. Traffic Flow Theory: a monograph. Special Report 165, Transportation Research Board (Washington D.C.: National Research Council) Gilchrist, R. 1988. Three-dimensional relationships in traffic flow theory variables. Unpublished Master's degree report, Dept. of Civil Engineering, McMaster University, Hamilton, Ontario, Canada. Gilchrist, R.S. and Hall, F.L. 1989. Three-dimensional relationships among traffic flow theory variables, Transportation Research Record 1225, 99-108. Greenberg, H. 1959. An analysis of traffic flow, Operations Research, Vol. 7, 78-85. Greenshields, B.D. 1935. A study of traffic capacity, HRB Proceedings 14, 448-477. Haight, F.A. 1960. The volume, density relation in the theory of road traffic, Operations Research 8, 572-573. Haight, F.A. 1963. Mathematical Theories of Traffic Flow. (New York: Academic Press) Hall, F.L. 1987. An interpretation of speed-flow-concentration relationships using catastrophe theory. Transportation Research A, 21A, 191-201. Hall, F.L., Allen, B.L. and Gunter, M.A. 1986. Empirical analysis of freeway flow-density relationships, Transportation Research A, 20A, 197-210. Hall, F.L. and Gunter, M.A. 1986. Further analysis of the flow-concentration relationship, Transportation Research Record 1091, 1-9. Hall, F.L. and Hall, L.M. 1990. Capacity and speed flow analysis of the QEW in Ontario" Transportation Research Record 1287, 108-118. Hall, F.L, Hurdle, V.F. and Banks, J.H. 1992. Synthesis of recent work on the nature of speed-flow and flow-occupancy (or density) relationships on freeways. In Transportation Research Record 1365, TRB, National Research Council, Washington, D.C. 12-18. Heidemann, D. and R. Hotop, R. 1990. Verteilung der Pkw-Geschwindigkeiten im Netz der


2-35

Bundesautobahnen- Modellmodifikation und Aktualisierung. Straße und Autobahn - Heft 2, 106-113. HCM 1965. Special Report 87: Highway Capacity Manual. Transportation Research Board. HCM 1985. Special Report 209: Highway Capacity Manual. Transportation Research Board. Hsu, P. and Banks, J.H. 1993. Effects of location on congested-regime flow-concentration relationships for freeways, Transportation Research Record 1398, 17-23. Huber, M.J. 1957. Effect of temporary bridge on parkway performance. Highway Research Board Bulletin 167 63-74. Hurdle, V.F. and Datta, P.K. 1983. Speeds and flows on an urban freeway: some measurements and a hypothesis. Transportation Research Record 905, 127-137. Koshi, M., Iwasaki, M. and Okhura, I. 1983. Some findings and an overview on vehicular flow characteristics. Proceedings, 8th International Symposium on Transportation and Traffic Flow Theory (Edited by Hurdle, V.F., Hauer, E. and Steuart, G.F.) University of Toronto Press, Toronto, Canada, 403-426. Kühne, R. and Immes, S. 1993. Freeway control systems for using section-related traffic variable detection. Proceedings of the Pacific Rim TransTech Conference, Seattle. New York: Am. Soc. Civil Eng., 56-62. Lighthill, M.J., and Whitham, G.B. 1955. On kinematic waves: II. A theory of traffic flow on long crowded roads. Proceedings of the Royal Society A229 No. 1178, 317-145. Makagami, Y., Newell, G.F. and Rothery, R. 1971. Three-dimensional representation of traffic flow, Transportation Science, Vol. 5, No. 3, pp. 302-313. May, A.D. 1990. Traffic Flow Fundamentals. Prentice-Hall. May, A.D., Athol, P., Parker, W. and Rudden, J.B. 1963. Development and evaluation of Congress Street Expressway pilot detection system, Highway Research Record 21, 48-70. Morton, T.W. and Jackson, C.P. 1992. Speed/flow geometry relationships for rural dual-carriageways and motorways. TRRL Contractor Report 279. 55 pages. Transport and Road Research Laboratory, Crowthorne, Berkshire. Moskowitz, K 1954 Waiting for a gap in a traffic stream Proc Highway Res Board 33, 385-395. Navin, F. 1986. Traffic congestion catastrophes. Transportation Planning Technology 11, 19-25.


2-36

Newell G.F. 1971 Applications of Queueing Theory. London: Chapman Hall. Newell, G.F. 1982. Applications of Queueing Theory, second edition. London: Chapman and Hall. Newell, G.F. 1993. A simplified theory of kinematic waves in highway traffic, Transportation Research B, 27B: Part I, General theory, 281-287; Part II, Queueing at freeway bottlenecks, 289-303; Part III, Multi-destination flows, 305-313. Newell, GF unpublished Notes on transportation operations Berkeley, CA: Univesity of California. Persaud, B.N. and Hall, F.L. 1989. Catastrophe theory and patterns in 30-second freeway data -- implications for incident detection, Transportation Research A 23A, 103-113. Persaud, B.N. and Hurdle, V.F. 1988. Some new data that challenge some old ideas about speed-flow relationships. Transportation Research Record 1194, 191-198. Ringert, J. and Urbanik, II, T. 1993. Study of freeway bottlenecks in Texas, Transportation Research Record 1398, 31-41. Robertson, D. 1994. Manual of Transportation Engineering Studies, fifth edition. Arlington, VA: Institute of Transportation Engineers. Stappert, K.H. and Theis, T.J. 1990. Aktualisierung der Verkehrsstärke- /Verkehrsgeschwindigkeitsbeziehungen des BVWP-Netzmodells. Research Report VU18009V89 Heusch/Boesefeldt, Aachen. Thom, R. 1975. Structural Stability and Morphogenesis. English translation by D.H. Fowler of Stabilite Structurelle et Morphogenese. Benjamin: Reading, Massachusetts. Underwood, R.T. 1961. Speed, volume, and density relationships: Quality and theory of traffic flow. Yale Bureau of Highway Traffic, pp. 141-188, as cited in Drake et al. 1967. Wardrop, J.G. and Charlesworth, G. 1954. A method of estimating speed and flow of traffic from a moving vehicle. Proceedings of the Institution of Civil Engineers, Part II, Vol. 3, 158-171. Wattleworth, J.A. 1963. Some aspects of macroscopic freeway traffic flow theory, Traffic Engineering 34: 2, 15-20. Wemple, E.A., Morris, A.M. and May, A.D. 1991. Freeway capacity and level of service concepts, in Highway Capacity and Level of Service, Proc. of the International Symposium on Highway Capacity, Karlsruhe (U. Brannolte, Ed.) Rotterdam: Balkema, 439-455. Wright, C. 1973. A theoretical analysis of the moving observer method, Transportation Research 7, 293-311.


2-37

Associate Professor, Texas A&M University, College Station, TX 77843.5

HUMAN FACTORS

BY RODGER J. KOPPA5


� = parameter of log normal distribution ~ standard deviation� = parameter of log normal distribution ~ median� = standard deviation� = value of standard normal variatea = maximum acceleration on gradeGV

a = maximum acceleration on levelLV

A = movement amplitudeC = roadway curvaturer

C = vehicle heading(t)

CV = coefficient of variationd = braking distanceD = distance from eye to targetE = symptom error function(t)

f = coefficient of frictionF = stability factors

g = acceleration of gravityg = control displacement(s)

G = gradientH = information (bits)K = gain (dB)l = wheel baseL = diameter of target (letter or symbol)LN = natural logM = meanMT = movement timeN = equiprobable alternativesPRT = perception-response timeR = desired input forcing function(t)

RT = reaction time (sec)s = Laplace operatorSR = steering ratio (gain)SSD = stopping sight distance t = timeT = lead term constantL

T = lag term constant�

T = neuro-muscular time constantN

u = speedV = initial speedW = width of control deviceZ = standard normal score

3 - 1

3.HUMAN FACTORS

3.1 Introduction In this chapter, salient performance aspects of the human in thecontext of a person-machine control system, the motor vehicle,will be summarized. The driver-vehicle system configuration isubiquitous. Practically all readers of this chapter are alsoparticipants in such a system; yet many questions, as will beseen, remain to be answered in modeling the behavior of thehuman component alone. Recent publications (IVHS 1992;TRB 1993) in support of Intelligent Transportation Systems(ITS) have identified study of "Plain Old Driving" (POD) as afundamental research topic in ITS. For the purposes of atransportation engineer interested in developing a molecularmodel of traffic flow in which the human in the vehicle or anindividual human-vehicle comprises a unit of analysis, someimportant performance characteristics can be identified to aid inthe formulation, even if a comprehensive transfer function for thedriver has not yet been formulated.

This chapter will proceed to describe first the discretecomponents of performance, largely centered aroundneuromuscular and cognitive time lags that are fundamentalparameters in human performance. These topics includeperception-reaction time, control movement time, responses tothe presentation of traffic control devices, responses to themovements of other vehicles, handling of hazards in theroadway, and finally how different segments of the drivingpopulation may differ in performance.

Next, the kind of control performance that underlies steering,braking, and speed control (the primary control functions) willbe described. Much research has focused on the development ofadequate models of the tracking behavior fundamental tosteering, much less so for braking or for speed control.

After fundamentals of open-loop and closed-loop vehicle controlare covered, applications of these principles to specificmaneuvers of interest to traffic flow modelers will be discussed.Lane keeping, car following, overtaking, gap acceptance, laneclosures, stopping and intersection sight distances will also bediscussed. To round out the chapter, a few other performanceaspects of the driver-vehicle system will be covered, such asspeed limit changes and distractions on the highway.

3.1.1 The Driving Task

Lunenfeld and Alexander (1990) consider the driving task to bea hierarchical process, with three levels: (1) Control,(2) Guidance, and (3) Navigation. The control level ofperformance comprises all those activities that involve second-to-second exchange of information and control inputs betweenthe driver and the vehicle. This level of performance is at thecontrol interface. Most control activities, it is pointed out, areperformed "automatically," with little conscious effort. In short,the control level of performance is skill based, in the approachto human performance and errors set forth by Jens Rasmussen aspresented in Human Error (Reason 1990).

Once a person has learned the rudiments of control of thevehicle, the next level of human performance in the driver-vehicle control hierarchy is the rules-based (Reason 1990)guidance level as Rasmussen would say. The driver's mainactivities "involve the maintenance of a safe speed and properpath relative to roadway and traffic elements ." (Lunenfeld andAlexander 1990) Guidance level inputs to the system aredynamic speed and path responses to roadway geometrics,hazards, traffic, and the physical environment. Informationpresented to the driver-vehicle system is from traffic controldevices, delineation, traffic and other features of theenvironment, continually changing as the vehicle moves alongthe highway.

These two levels of vehicle control, control and guidance, are ofparamount concern to modeling a corridor or facility. The third(and highest) level in which the driver acts as a supervisor apart,is navigation. Route planning and guidance while enroute, forexample, correlating directions from a map with guide signagein a corridor, characterize the navigation level of performance.Rasmussen would call this level knowledge-based behavior.Knowledge based behavior will become increasingly moreimportant to traffic flow theorists as Intelligent TransportationSystems (ITS) mature. Little is currently known about howenroute diversion and route changes brought about by ITStechnology affect traffic flow, but much research is underway.This chapter will discuss driver performance in the conventionalhighway system context, recognizing that emerging ITStechnology in the next ten years may radically change manydriver's roles as players in advanced transportation systems.

3. HUMAN FACTORS

3 - 2

At the control and guidance levels of operation, the driver of a driver-vehicle system from other vehicles, the roadway, and themotor vehicle has gradually moved from a significant primemover, a supplier of forces to change the path of the vehicle, toan information processor in which strength is of little or noconsequence. The advent of power assists and automatictransmissions in the 1940's, and cruise controls in the 1950'smoved the driver more to the status of a manager in the system.There are commercially available adaptive controls for severelydisabled drivers (Koppa 1990) which reduce the actualmovements and strength required of drivers to nearly thevanishing point. The fundamental control tasks, however,remain the same.

These tasks are well captured in a block diagram first developedmany years ago by Weir (1976). This diagram, reproduced inFigure 3.1, forms the basis for the discussion of driverperformance, both discrete and continuous. Inputs enter the

driver him/herself (acting at the navigation level ofperformance).

The fundamental display for the driver is the visual field as seenthrough the windshield, and the dynamics of changes to that fieldgenerated by the motion of the vehicle. The driver attends toselected parts of this input, as the field is interpreted as the visualworld. The driver as system manager as well as active systemcomponent "hovers" over the control level of performance.Factors such as his or her experience, state of mind, andstressors (e.g., being on a crowded facility when 30 minutes late for a meeting) all impinge on the supervisory ormonitoring level of performance, and directly or indirectly affectthe control level of performance. Rules and knowledge governdriver decision making and the second by second psychomotoractivity of the driver. The actual control

Figure 3.1Generalized Block Diagram of the Car-Driver-Roadway System.

3. HUMAN FACTORS

3 - 3

movements made by the driver couple with the vehicle control As will be discussed, a considerable amount of information isat the interface of throttle, brake, and steering. The vehicle, in available for some of the lower blocks in this diagram, the onesturn, as a dynamic physical process in its own right, is subject to associated with braking reactions, steering inputs, andvehic leinputs from the road and the environment. The resolution of control dynamics. Far less is really known about the higher-control dynamics and vehicle disturbance dynamics is the vehicle order functions that any driver knows are going on while he orpath. she drives.

3.2 Discrete Driver Performance

3.2.1 Perception-Response Time

Nothing in the physical universe happens instantaneously. Underlying the Hick-Hyman Law is the two-component concept:Compared to some physical or chemical processes, the simplest part of the total time depends upon choice variables, and part ishuman reaction to incoming information is very slow indeed. common to all reactions (the intercept). Other components canEver since the Dutch physiologist Donders started to speculate be postulated to intervene in the choice variable component,in the mid 19th century about central processes involved in other than just the information content. Most of these modelschoice and recognition reaction times, there have been numerousmodels of this process. The early 1950's saw InformationTheory take a dominant role in experimental psychology. Thelinear equation

RT = a + bH (3.1)

Where: the 85th percentile estimate for that aspect of time lag. Because

RT = Reaction time, secondsH = Estimate of transmitted informationH = log N , if N equiprobable alternatives2

a = Minimum reaction time for that modalityb = Empirically derived slope, around 0.13

seconds (sec) for many performance situations

that has come to be known as the Hick-Hyman "Law" expressesa relationship between the number of alternatives that must besorted out to decide on a response and the total reaction time,that is, that lag in time between detection of an input (stimulus)and the start of initiation of a control or other response. If thetime for the response itself is also included, then the total lag istermed "response time." Often, the terms "reaction time" and"response time" are used interchangeably, but one (reaction) isalways a part of the other (response).

have then been chaining individual components that arepresumably orthogonal or uncorrelated with one another.Hooper and McGee (1983) postulate a very typical and plausiblemodel with such components for braking response time,illustrated in Table 3.1.

Each of these elements is derived from empirical data, and is in

it is doubtful that any driver would produce 85th percentilevalues for each of the individual elements, 1.50 secondsprobably represents an extreme upper limit for a driver'sperception-reaction time. This is an estimate for the simplestkind of reaction time, with little or no decision making. Thedriver reacts to the input by lifting his or her foot from theaccelerator and placing it on the brake pedal. But a number ofwriters, for example Neuman (1989), have proposed perception-reaction times (PRT) for different types of roadways, rangingfrom 1.5 seconds for low-volume roadways to 3.0 seconds forurban freeways. There are more things happening, and moredecisions to be made per unit block of time on a busy urbanfacility than on a rural county road. Each of those added factorsincrease the PRT. McGee (1989) has similarly proposeddifferent values of PRT as a function of design speed. Theseestimates, like those in Table 3.1, typically include the time forthe driver to move his or her foot from the accelerator to thebrake pedal for brake application.

3. HUMAN FACTORS

3 - 4

Table 3.1Hooper-McGee Chaining Model of Perception-Response Time

Component (sec) (sec)Time Cumulative Time

1) Perception

Latency 0.31 0.31

Eye Movement 0.09 0.4

Fixation 0.2 1

Recognition 0.5 1.5

2) Initiating Brake 1.24 2.74 Application

Any statistical treatment of empirically obtained PRT's should skew, because there cannot be such a thing as a negative reactiontake into account a fundamental if not always vitally important time, if the time starts with onset of the signal with nofact: the times cannot be distributed according to the normal or anticipation by the driver. Taoka (1989) has suggested angaussian probability course. Figure 3.2 illustrates the actual adjustment to be applied to PRT data to correct for the non-shape of the distribution. The distribution has a marked positive normality, when sample sizes are "large" --50 or greater.

Figure 3.2Lognormal Distribution of Perception-Reaction Time.

f(t) � 12� �t

exp LN(t)��

2

�2� LN 1� �2

μ2

� � LN μ

1��2/μ2

�LN(t)��

�� 0.5, 0.85, etc.

LN(t)��

� Z

3. HUMAN FACTORS

3 - 5

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

The log-normal probability density function is widely used inquality control engineering and other applications in whichvalues of the observed variable, t, are constrained to values equalto or greater than zero, but may take on extreme positive values,exactly the situation that obtains in considering PRT. In suchsituations, the natural logarithm of such data may be assumed toapproach the normal or gaussian distribution. Probabilitiesassociated with the log-normal distribution can thus be Therefore, the value of LN(t) for such percentile levels as 0.50determined by the use of standard-score tables. Ang and Tang (the median), the 85th, 95th, and 99th can be obtained by(1975) express the log-normal probability density function f(t) substituting in Equation 3.6 the appropriate Z score of 0.00,as follows:

where the two parameters that define the shape of thedistribution are � and �. It can be shown that these twoparameters are related to the mean and the standard deviation ofa sample of data such as PRT as follows:

The parameter � is related to the median of the distribution beingdescribed by the simple relationship of the natural logarithm ofthe median. It can also be shown that the value of the standardnormal variate (equal to probability) is related to theseparameters as shown in the following equation:

and the standard score associated with that value is given by:

1.04, 1.65, and 2.33 for Z and then solving for t. Convertingdata to log-normal approximations of percentile values should beconsidered when the number of observations is reasonably large,over 50 or more, to obtain a better fit. Smaller data sets willbenefit more from a tolerance interval approach to approximatepercentiles (Odeh 1980).

A very recent literature review by Lerner and his associates(1995) includes a summary of brake PRT (including brakeonset) from a wide variety of studies. Two types of responsesituation were summarized: (1) The driver does not know whenor even if the stimulus for braking will occur, i.e., he or she issurprised, something like a real-world occurrence on thehighway; and (2) the driver is aware that the signal to brake willoccur, and the only question is when. The Lerner et al. (1995)composite data were converted by this writer to a log-normaltransformation to produce the accompanying Table 3.2.

Sixteen studies of braking PRT form the basis for Table 3.2.Note that the 95th percentile value for a "surprise" PRT (2.45seconds) is very close to the AASHTO estimate of 2.5 secondswhich is used for all highway situations in estimating bothstopping sight distance and other kinds of sight distance (Lerneret al. 1995).

In a very widely quoted study by Johansson and Rumar (1971),drivers were waylaid and asked to brake very briefly if theyheard a horn at the side of the highway in the next 10 kilometers.Mean PRT for 322 drivers in this situation was 0.75 secondswith an SD of 0.28 seconds. Applying the Taoka conversion tothe log normal distribution yields:

50th percentile PRT = 0.84 sec85th percentile PRT = 1.02 sec95th percentile PRT = 1.27 sec99th percentile PRT = 1.71 sec

3. HUMAN FACTORS

3 - 6

Table 3.2Brake PRT - Log Normal Transformation

"Surprise" "Expected"

Mean 1.31 (sec) 0.54

Standard Dev 0.61 0.1

� 0.17 (no unit) -0.63 (no unit)

� 0.44 (no unit) 0.18 (no unit)

50th percentile 1.18 0.53




In very recent work by Fambro et al. (1994) volunteer drivers in Additional runs were made with other drivers in their own carstwo age groups (Older: 55 and up; and Young: 18 to 25) were equipped with the same instrumentation. Nine of the 12 driverssuddenly presented with a barrier that sprang up from a slot in made stopping maneuvers in response to the emergence of thethe pavement in their path, with no previous instruction. They barrier. The results are given in Table 3.3 as Case 2. In anwere driving a test vehicle on a closed course. Not all 26 drivers attempt (Case 3) to approximate real-world driving conditions,hit the brakes in response to this breakaway barrier. The PRT's Fambro et al. (1994) equipped 12 driver's own vehicles withof the 22 who did are summarized in Table 3.3 (Case 1). None instrumentation. They were asked to drive a two-lane undividedof the age differences were statistically significant. secondary road ostensibly to evaluate the drivability of the road.

Table 3.3Summary of PRT to Emergence of Barrier or Obstacle

Case 1. Closed Course, Test Vehicle

12 Older: Mean = 0.82 sec; SD = 0.16 sec

10 Young: Mean = 0.82 sec; SD = 0.20 sec

Case 2. Closed Course, Own Vehicle

7 Older: Mean = 1.14 sec; SD = 0.35 sec

3 Young: Mean = 0.93 sec; SD = 0.19 sec

Case 3. Open Road, Own Vehicle

5 Older: Mean = 1.06 sec; SD = 0.22 sec6 Young: Mean = 1.14 sec; SD = 0.20 sec

3. HUMAN FACTORS

3 - 7

A braking incident was staged at some point during this test event to an expected event ranges from 1.35 to 1.80 sec,drive. A barrel suddenly rolled out of the back of a pickup consistent with Johansson and Rumar (1971). Note, however,parked at the side of the road as he or she drove by. The barrel that one out of 12 of the drivers in the open road barrel studywas snubbed to prevent it from actually intersecting the driver'spath, but the driver did not know this. The PRT's obtained bythis ruse are summarized in Table 3.4. One driver failed tonotice the barrel, or at least made no attempt to stop or avoid it.

Since the sample sizes in these last two studies were small, itwas considered prudent to apply statistical tolerance intervals tothese data in order to estimate proportions of the drivingpopulation that might exhibit such performance, rather thanusing the Taoka conversion. One-sided tolerance tablespublished by Odeh (1980) were used to estimate the percentageof drivers who would respond in a given time or shorter, basedon these findings. These estimates are given in Table 3.4 (95percent confidence level), with PRT for older and youngerdrivers combined.

The same researchers also conducted studies of driver responseto expected obstacles. The ratio of PRT to a totally unexpected

(Case 3) did not appear to notice the hazard at all. Thirtypercent of the drivers confronted by the artificial barrier underclosed-course conditions also did not respond appropriately.How generalizable these percentages are to the driver populationremains an open question that requires more research. Foranalysis purposes, the values in Table 3.4 can be used toapproximate the driver PRT envelope for an unexpected event.PRT's for expected events, e.g., braking in a queue in heavytraffic, would range from 1.06 to 1.41 second, according to theratios given above (99th percentile).

These estimates may not adequately characterize PRT underconditions of complete surprise, i.e., when expectancies aregreatly violated (Lunenfeld and Alexander 1990). Detectiontimes may be greatly increased if, for example, an unlightedvehicle is suddenly encountered in a traffic lane in the dark, tosay nothing of a cow or a refrigerator.

Table 3.4Percentile Estimates of PRT to an Unexpected Object

Percentile Closed Course Closed Course Open Road

Case 1 Case 2 Case 3Test Vehicle Own Vehicle Own Vehicle

50th 0.82 sec 1.09 sec 1.11 sec

75th 1.02 sec 1.54 sec 1.40 sec

90th 1.15 sec 1.81 sec 1.57 sec

95th 1.23 sec 1.98 sec 1.68 sec

99th 1.39 sec 2.31 sec 1.90 sec

Adapted from Fambro et al. (1994).

MT � a�b Log22AW

Log22AW

MT � a�b A

3. HUMAN FACTORS

3 - 8

(3.7)

(3.8)

(3.9)

3.3 Control Movement Time

Once the lag associated with perception and then reaction has is the "Index of Difficulty" of the movement, in binary units, thusensued and the driver just begins to move his or her foot (or linking this simple relationship with the Hick-Hyman equationhand, depending upon the control input to be effected), the discussed previously in Section 3.3.1. amount of time required to make that movement may be ofinterest. Such control inputs are overt motions of an appendage Other researchers, as summarized by Berman (1994), soonof the human body, with attendant inertia and muscle fiber found that certain control movements could not be easilylatencies that come into play once the efferent nervous impulses modeled by Fitts' Law. Accurate tapping responses less thanarrives from the central nervous system. 180 msec were not included. Movements which are short and

3.3.1 Braking Inputs

As discussed in Section 3.3.1 above, a driver's braking responseis composed of two parts, prior to the actual braking of thevehicle: the perception-reaction time (PRT) and immediatelyfollowing, movement time (MT ).

Movement time for any sort of response was first modeled byFitts in 1954. The simple relationship among the range oramplitude of movement, size of the control at which the controlmovement terminates, and basic information about the minimum"twitch" possible for a control movement has long been knownas "Fitts' Law."

where,a = minimum response time lag, no movementb = slope, empirically determined, different for

each limbA = amplitude of movement, i.e., the distance

from starting point to end pointW = width of control device (in direction of

movement)

The term

quick also appear to be preplanned, or "programmed," and areopen-loop. Such movements, usually not involving visualfeedback, came to be modeled by a variant of Fitts' Law:

in which the width of the target control (W ) plays no part.

Almost all such research was devoted to hand or arm responses.In 1975, Drury was one of the first researchers to test theapplicability of Fitts' Law and its variants to foot and legmovements. He found a remarkably high association for fittingfoot tapping performance to Fitts' Law. Apparently, allappendages of the human body can be modeled by Fitts' Law orone of its variants, with an appropriate adjustment of a and b, theempirically derived parameters. Parameters a and b aresensitive to age, condition of the driver, and circumstances suchas degree of workload, perceived hazard or time stress, and pre-programming by the driver.

In a study of pedal separation and vertical spacing between theplanes of the accelerator and brake pedals, Brackett and Koppa(1988) found separations of 10 to 15 centimeters (cm), with littleor no difference in vertical spacing, producedcontrol movement in the range of 0.15 to 0.17 sec. Raising thebrake pedal more than 5 cm above the accelerator lengthenedthis time significantly. If pedal separation ( = A in Fitts' Law)was varied, holding pedal size constant, the mean MT was 0.22sec, with a standard deviation of 0.20 sec.

In 1991, Hoffman put together much of the extant literature andconducted studies of his own. He found that the Index ofDifficulty was sufficiently low (<1.5) for all pedal placementsfound on passenger motor vehicles that visual control was

3. HUMAN FACTORS

3 - 9

unnecessary for accurate movement, i.e., movements were other relationship between these two times. A recent analysis byballistic in nature. MT was found to be greatly influenced byvertical separation of the pedals, but comparatively little bychanges in A, presumably because the movements were ballisticor open-loop and thus not correctable during the course of themovement. MT was lowest at 0.20 sec with no verticalseparation, and rose to 0.26 sec if the vertical separation (brakepedal higher than accelerator) was as much as 7 cm. A veryrecent study by Berman (1994) tends to confirm these generalMT evaluations, but adds some additional support for a ballisticmodel in which amplitude A does make a difference.

Her MT findings for a displacement of (original) 16.5 cm and(extended) 24.0 cm, or change of 7.5 cm can be summarized asfollows:

Original pedal Extended pedalMean 0.20 sec 0.29 secStandard Deviation 0.05 sec 0.07 sec95 percent tolerance level 0.32 sec 0.45 sec99 percent tolerance level 0.36 sec 0.51 sec

The relationship between perception-reaction time and MT hasbeen shown to be very weak to nonexistent. That is, a longreaction time does not necessarily predict a long MT, or any

the writer yielded a Pearson Product-Moment CorrelationCoefficient (r) value of 0.17 between these two quantities in abraking maneuver to a completely unexpected object in the pathof the vehicle (based on 21 subjects). Total PRT's as presentedin Section 3.3.1 should be used for discrete braking controlmovement time estimates; for other situations, the modeler coulduse the tolerance levels in Table 3.5 for MT, chaining them afteran estimate of perception (including decision) and reaction timefor the situation under study (95 percent confidence level). SeeSection 3.14 for a discussion on how to combine these estimates.

3.3.2 Steering Response Times

Summala (1981) covertly studied driver responses to the suddenopening of a car door in their path of travel. By "covert" ismeant the drivers had no idea they were being observed or wereparticipating in the study. This researcher found that neither thelatency nor the amount of deviation from the pre-event pathwaywas dependent upon the car's prior position with respect to theopening car door. Drivers responded with a ballistic "jerk" ofthe steering wheel. The mean response latency for these Finnishdrivers was 1.5 sec, and reached the half-way point of maximumdisplacement from the original path in about 2.5 sec. The

Table 3.5Movement Time Estimates

Source N Mean 75th 90th 95th 99th(Std) Sec Sec Sec Sec

Brackett (Brackett and Koppa 1988) 24 0.22 (0.20) 0.44 0.59 0.68 0.86

Hoffman (1991) 18 0.26 (0.20) 0.50 0.66 0.84 1.06

Berman (1994) 24 0.20 (0.05) 0.26 0.29 0.32 0.36

3. HUMAN FACTORS

3 - 10

3.4 Response Distances and Times to Traffic Control DevicesThe driving task is overwhelmingly visual in nature; external very rich with data related to target detection in complex visualinformation coming through the windshield constitutes nearly all environments, and a TCD's target value also depends upon thethe information processed. A major input to the driver which driver's predilection to look for and use such devices. influences his or her path and thus is important to traffic flowtheorists is traffic information imparted by traffic control devices(TCD). The major issues concerned with TCD are all related todistances at which they may be (1) detected as objects in thevisual field; (2) recognized as traffic control devices: signs,signals, delineators, and barricades; (3) legible or identifiable sothat they may be comprehended and acted upon. Figure 3.3depicts a conceptual model for TCD information processing, andthe many variables which affect it. The research literature is

3.4.1 Traffic Signal Change

From the standpoint of traffic flow theory and modeling, a majorconcern is at the stage of legibility or identification and acombination of "read" and "understand" in the diagram in Figure3.3. One of the most basic concerns is driver response or lag to

Figure 3.3A Model of Traffic Control Device Information Processing.

3. HUMAN FACTORS

3 - 11

changing traffic signals. Chang et al. (1985) found through Mean PRT to signal change = 1.30 seccovert observations at signalized intersections that drivers 85th percentile PRT = 1.50 secresponse lag to signal change (time of change to onset of brake 95th percentile PRT = 2.50 seclamps) averaged 1.3 sec, with the 85th percentile PRT estimated 99th percentile PRT = 2.80 secat 1.9 sec and the 95th percentile at 2.5 sec. This PRT to signalchange is somewhat inelastic with respect to distance from the If the driver is stopped at a signal, and a straight-ahead maneuvertraffic signal at which the signal state changed. The mean PRT is planned, PRT would be consistent with those values given in(at 64 kilometers per hour (km/h)) varied by only 0.20 sec within Section 3.3.1. If complex maneuvers occur after signal changea distance of 15 meters (m) and by only 0.40 sec within 46 m. (e.g., left turn yield to oncoming traffic), the Hick-Hyman Law

Wortman and Matthias (1983) found similar results to Chang etal. (1985) with a mean PRT of 1.30 sec, and a 85th percentilePRT of 1.5 sec. Using tolerance estimates based on theirsample size, (95 percent confidence level) the 95th percentilePRT was 2.34 sec, and the 99th percentile PRT was 2.77 sec.They found very little relationship between the distance from theintersection and either PRT or approach speed (r = 0.08). So2

the two study findings are in generally good agreement, and thefollowing estimates may be used for driver response to signalchange:

(Section 3.2.1) could be used with the y intercept being the basicPRT to onset of the traffic signal change. Considerations relatedto intersection sight distances and gap acceptance make suchpredictions rather difficult to make without empirical validation.These considerations will be discussed in Section 3.15.

3.4.2 Sign Visibility and Legibility

The psychophysical limits to legibility (alpha-numeric) andidentification (symbolic) sign legends are the resolving power of

Table 3.6Visual Acuity and Letter Sizes

Snellen Acuity Visual angle of letter or symbol Legibility Index

SI (English) 'of arc radians m/cm

6/3 (20/10) 2.5 0.00073 13.7

6/6 (20/20) 5 0.00145 6.9

6/9 (20/30) 7.5 0.00218 4.6

6/12 (20/40) 10 0.00291 3.4

6/15 (20/50) 12.5 0.00364 2.7

6/18 (20/60) 15 0.00436 2.3

� � 2 arctan L2D

� � 2 arctan L2D

3. HUMAN FACTORS

3 - 12

(3.10)(3.10)

the visual perception system, the effects of the optical train confirmed these earlier findings, and also notes that extremeleading to presentation of an image on the retina of the eye,neural processing of that image, and further processing by thebrain. Table 3.6 summarizes visual acuity in terms of visualangles and legibility indices.

The exact formula for calculating visual angle is

where, L = diameter of the target (letter or symbol)D = distance from eye to target in the same units

All things being equal, two objects that subtend the same visualangle will elicit the same response from a human observer,regardless of their actual sizes and distances. In Table 3.6 theSnellen eye chart visual acuity ratings are related to the size ofobjects in terms of visual arc, radians (equivalent for small sizesto the tangent of the visual arc) and legibility indices. Standardtransportation engineering resources such as the Traffic Control depending upon sign complexity. These differences consistentDevices Handbook (FHWA 1983) are based upon thesefundamental facts about visual performance, but it should beclearly recognized that it is very misleading to extrapolatedirectly from letter or symbol legibility/recognition sizes to signperceptual distances, especially for word signs. There are otherexpectancy cues available to the driver, word length, layout, etc.that can lead to performance better than straight visual anglecomputations would suggest. Jacobs, Johnston, and Cole (1975)also point out an elementary fact that 27 to 30 percent of thedriving population cannot meet a 6/6 (20/20) criterion. Moststates in the U.S. have a 6/12 (20/40) static acuity criterion forunrestricted licensure, and accept 6/18 (20/60) for restricted(daytime, usually) licensure. Such tests in driver license officesare subject to error, and examiners tend to be very lenient.Night-time static visual acuity tends to be at least one Snellenline worse than daytime, and much worse for older drivers (to bediscussed in Section 3.8).

Jacobs, et al. also point out that the sign size for 95th percentilerecognition or legibility is 1.7 times the size for 50th percentileperformance. There is also a pervasive notion in the researchthat letter sign legibility distances are half symbol signrecognition distances, when drivers are very familiar with thesymbol (Greene 1994). Greene (1994), in a very recent study,

variability exists from trial to trial for the same observer on a given sign's recognition distance. Presumably, word signs wouldmanifest as much or even more variability. Complex, fine detailsigns such as Bicycle Crossing (MUTCD W11-1) wereobserved to have coefficients of variation between subjects of 43percent. Coefficient of Variation (CV) is simply:

CV = 100 � (Std Deviation/Mean) (3.11)

In contrast, very simple symbol signs such as T-Junction(MUTCD W2-4) had a CV of 28 percent. Within subjectvariation (from trial to trial) on the same symbol sign issummarized in Table 3.7.

Before any reliable predictions can be made about legibility orrecognition distances of a given sign, Greene (1994) found thatsix or more trials under controlled conditions must be made,either in the laboratory or under field conditions. Greene (1994)found percent differences between high-fidelity laboratory andfield recognition distances to range from 3 to 21 percent,

with most researchers, were all in the direction of laboratorydistances being greater than actual distances; the laboratorytends to overestimate field legibility distances. Variability inlegibility distances, however, is as great in the laboratory as it isunder field trials.

With respect to visual angle required for recognition, Greenefound, for example, that the Deer Crossing at the meanrecognition distance had a mean visual angle of 0.00193 radian,or 6.6 minutes of arc. A more complex, fine detail sign such asBicycle Crossing required a mean visual angle of 0.00345 radianor 11.8 minutes of arc to become recognizable.

With these considerations in mind, here is the bestrecommendation that this writer can make. For the purposes ofpredicting driver comprehension of signs and other devices thatrequire interpretation of words or symbols use the data in Table3.6 as "best case," with actual performance expected to besomewhat to much worse (i.e., requiring closer distances for agiven size of character or symbol). The best visual acuity thatcan be expected of drivers under optimum contrast conditions sofar as static acuity is concerned would be 6/15 (20/50) when thesizable numbers of older drivers is considered [13 percent in1990 were 65 or older (O'Leary and Atkins 1993)].

3. HUMAN FACTORS

3 - 13

Table 3.7Within Subject Variation for Sign Legibility

Young Drivers Older Drivers

Sign Min CV Max CV Min CV Max CV

WG-3 2 Way Traffic 3.9 21.9 8.9 26.7

W11-1 Bicycle Cross 6.7 37.0 5.5 39.4

W2-1 Crossroad 5.2 16.3 2.0 28.6

W11-3 Deer Cross 5.4 21.3 5.4 49.2

W8-5 Slippery 7.7 33.4 15.9 44.1

W2-5 T-Junction 5.6 24.6 4.9 28.7

3.4.3 Real-Time Displays and Signs

With the advent of Intelligent Transportation Systems (ITS),traffic flow modelers must consider the effects of changeablemessage signs on driver performance in traffic streams.Depending on the design of such signs, visual performance tothem may not differ significantly from conventional signage.Signs with active (lamp or fiber optic) elements may not yieldthe legibility distances associated with static signage, because Federal Highway Administration, notably the definitive manualby Dudek (1990).

3.4.4 Reading Time Allowance

For signs that cannot be comprehended in one glance, i.e., wordmessage signs, allowance must be made for reading theinformation and then deciding what to do, before a driver intraffic will begin to maneuver in response to the information.Reading speed is affected by a host of factors (Boff and Lincoln1988) such as the type of text, number of words, sentencestructure, information order, whatever else the driver is doing,the purpose of reading, and the method of presentation. TheUSAF resource (Boff and Lincoln 1988) has a great deal ofgeneral information on various aspects of reading sign material.For purposes of traffic flow modeling, however, a general rule ofthumb may suffice. This can be found in Dudek (1990):

"Research...has indicated that a minimum exposure time of onesecond per short word (four to eight characters) (exclusive ofprepositions and other similar connectors) or two seconds perunit of information, whichever is largest, should be used forunfamiliar drivers. On a sign having 12 to 16 characters perline, this minimum exposure time will be two seconds per line.""Exposure time" can also be interpreted as "reading time" and soused in estimating how long drivers will take to read andcomprehend a sign with a given message.

Suppose a sign reads:

Traffic ConditionsNext 2 Miles

Disabled Vehicle on I-77Use I-77 Bypass Next Exit

Drivers not familiar with such a sign ("worst case," but able toread the sign) could take at least 8 seconds and according to theDudek formula above up to 12 seconds to process thisinformation and begin to respond. In Dudek's 1990 study, 85percent of drivers familiar with similar signs read this 13-wordmessage (excluding prepositions) with 6 message units in 6.7seconds. The formulas in the literature properly tend to beconservative.

3. HUMAN FACTORS

3 - 14

3.5 Response to Other Vehicle Dynamics

Vehicles in a traffic stream are discrete elements with motion ahead) is the symmetrical magnification of a form or texture incharacteristics loosely coupled with each other via the driver's the field of view. Visual angle transitions from a near-linear toprocessing of information and making control inputs. Effects of a geometric change in magnitude as an object approaches atchanges in speed or acceleration of other elements as perceived constant velocity, as Figure 3.4 depicts for a motor vehicleand acted on by the driver of any given element are of interest. approaching at a delta speed of 88 km/h. As the rate of changeTwo situations appear relevant: (1) the vehicle ahead and (2) the of visual angle becomes geometric, the perceptual systemvehicle alongside (in the periphery). triggers a warning that an object is going to collide with the

3.5.1 The Vehicle Ahead

Consideration of the vehicle ahead has its basis in thresholds fordetection of radial motion (Schiff 1980). Radial motion ischange in the apparent size of a target. The minimum conditionfor perceiving radial motion of an object (such as a vehicle

observer, or, conversely, that the object is pulling away from theobserver. This phenomenon is called looming. If the rate ofchange of visual angle is irregular, that is information to theperceptual system that the object in motion is moving at achanging velocity (Schiff 1980). Sekuler and Blake (1990)report evidence that actual looming detectors exist in the humanvisual system. The relative change in visual angle is roughlyequal to the reciprocal of "time-to-go" (time to impact), a special

Figure 3.4Looming as a Function of Distance from Object.

3. HUMAN FACTORS

3 - 15

case of the well-known Weber fraction, S = �I/I, the magnitudeof a stimulus is directly related to a change in physical energy butinversely related to the initial level of energy in the stimulus.

Human visual perception of acceleration (as such) of an objectin motion is very gross and inaccurate; it is very difficult for adriver to discriminate acceleration from constant velocity unlessthe object is observed for a relatively long period of time - 10 or15 sec (Boff and Lincoln 1988).

The delta speed threshold for detection of oncoming collision orpull-away has been studied in collision-avoidance research.Mortimer (1988) estimates that drivers can detect a change indistance between the vehicle they are driving and the one in frontwhen it has varied by approximately 12 percent. If a driver werefollowing a car ahead at a distance of 30 m, at a change of 3.7 mthe driver would become aware that distance is decreasing orincreasing, i.e., a change in relative velocity. Mortimer notesthat the major cue is rate of change in visual angle. Thisthreshold was estimated in one study as 0.0035 radians/sec.

This would suggest that a change of distance of 12 percent in 5.6seconds or less would trigger a perception of approach or pullingaway. Mortimer concludes that "...unless the relative velocitybetween two vehicles becomes quite high, the drivers will

respond to changes in their headway, or the change in angularsize of the vehicle ahead, and use that as a cue to determine thespeed that they should adopt when following another vehicle."

3.5.2 The Vehicle Alongside

Motion detection in peripheral vision is generally less acute thanin foveal (straight-ahead) vision (Boff and Lincoln 1988), in thata greater relative velocity is necessary for a driver "looking outof the corner of his eye" to detect that speedchange than if he or she is looking to the side at the subjectvehicle in the next lane. On the other hand, peripheral vision isvery blurred and motion is a much more salient cue than astationary target is. A stationary object in the periphery (such asa neighboring vehicle exactly keeping pace with the driver'svehicle) tends to disappear for all intents and purposes unless itmoves with respect to the viewer against a patterned background.Then that movement will be detected. Relative motion in theperiphery also tends to look slower than the same movement asseen using fovea vision. Radial motion (car alongside swervingtoward or away from the driver) detection presumably wouldfollow the same pattern as the vehicle ahead case, but no studyconcerned with measuring this threshold directly was found.

3.6 Obstacle and Hazard Detection, Recognition, and Identification

Drivers on a highway can be confronted by a number of different roadway were unexpectedly encountered by drivers on a closedsituations which dictate either evasive maneuvers or stopping course. Six objects, a 1 x 4 board, a black toy dog, a white toymaneuvers. Perception-response time (PRT) to such encounters dog, a tire tread, a tree limb with leaves, and a hay bale werehave already been discussed in Section 3.3.1. But before a placed in the driver's way. Both detection and recognitionmaneuver can be initiated, the object or hazard must first be distances were recorded. Average visual angles of detection fordetected and then recognized as a hazard. The basic these various objects varied from the black dog at 1.8 minimumconsiderations are not greatly different than those discussed under of arc to 4.9 min of arc for the tree limb. Table 3.8 summarizesdriver responses to traffic control devices (Section 3.5), but some the detection findings of this study.specific findings on roadway obstacles and hazards will also bediscussed. At the 95 percent level of confidence, it can be said from these

3.6.1 Obstacle and Hazard Detection

Picha (1992) conducted an object detection study in whichrepresentative obstacles or objects that might be found on a

findings that an object subtending a little less than 5 minutes ofarc will be detected by all but 1 percent of drivers under daylightconditions provided they are looking in the object's direction.Since visual acuity declines by as much as two Snellen lines afternightfall, to be dtected such targets with similar contrast would

3. HUMAN FACTORS

3 - 16

Table 3.8Object Detection Visual Angles (Daytime)(Minutes of Arc)

Object MeanTolerance, 95th confidence

STD 95th 99th

1" x 4" Board, 24" x 1"* 2.47 1.21 5.22 6.26

Black toy dog, 6" x 6" 1.81 0.37 2.61 2.91

White toy dog, 6" x 6" 2.13 0.87 4.10 4.84

Tire tread, 8" x 18" 2.15 0.38 2.95 3.26

Tree Branch, 18" x 12" 4.91 1.27 7.63 8.67

Hay bale, 48" x 18" 4.50 1.28 7.22 8.26

All Targets 3.10 0.57 4.30 4.76

*frontal viewing plan dimensions

have to subtend somewhere around 2.5 times the visual angle that 15 cm or less in height very seldom are causal factors in accidentsthey would at detection under daylight conditions. (Kroemer et al. 1994).

3.6.2 Obstacle and Hazard Recognition and Identification

Once the driver has detected an object in his or her path, the nextjob is to: (1) decide if the object, whatever it is, is a potentialhazard, this is the recognition stage, followed by (2) theidentification stage, even closer, at which a driver actually can tellwhat the object is. If an object (assume it is stationary) is smallenough to pass under the vehicle and between the wheels, itdoesn't matter very much what it is. So the first estimate isprimarily of size of the object. If the decision is made that theobject is too large to pass under the vehicle, then either evasiveaction or a braking maneuver must be decided upon. Objects

The majority of objects encountered on the highway thatconstitute hazard and thus trigger avoidance maneuvers are largerthan 60 cm in height. Where it may be of interest to establish avisual angle for an object to be discriminated as a hazard or non-hazard, such decisions require visual angles on the order of atleast the visual angles identified in Section 3.4.2 for letter orsymbol recognition, i.e., about 15 minutes of arc to take in 99percent of the driver population. It would be useful to reflect thatthe full moon subtends 30 minutes of arc, to give the reader anintuitive feel for what the minimum visual angle might be forobject recognition. At a distance somewhat greater than this, thedriver decides if an object is sizable enough to constitute a hazard,largely based upon roadway lane width size comparisons and thesize of the object with respect to other familiar roadside objects(such as mailboxes, bridge rails). Such judgements improve ifthe object is identified.

3. HUMAN FACTORS

3 - 17

3.7 Individual Differences in Driver Performance

In psychological circles, variability among people, especially that visual and cognitive changes affecting driver performance willassociated with variables such as gender, age, socio-economic be discussed in the following paragraphs.levels, education, state of health, ethnicity, etc., goes by the name"individual differences." Only a few such variables are of interestto traffic flow modeling. These are the variables which directlyaffect the path and velocity the driven vehicle follows in a giventime in the operational environment. Other driver characteristicswhich may be of interest to the reader may be found in the NHTSA than 20/50 corrected, owing to senile macular degenerationDriver Performance Data book (1987, 1994).

3.7.1 Gender

Kroemer, Kroemer, and Kroemer-Ebert (1994) summarizerelevant gender differences as minimal to none. Fine fingerdexterity and color perception are areas in which women performbetter than men, but men have an advantage in speed. Reactiontime tends to be slightly longer for women than for men the recentpopular book and PBS series, Brain Sex (Moir and Jessel 1991)has some fascinating insights into why this might be so. Thisdifference is statistically but not practically significant. For thepurpose of traffic flow analysis, performance differences betweenmen and women may be ignored.

3.7.2 Age

Research on the older driver has been increasing at an exponentialrate, as was noted in the recent state-of-the-art summary by theTransportation Research Board (TRB 1988). Although a numberof aspects of human performance related to driving change withthe passage of years, such as response time, channel capacity andprocessing time needed for decision making, movement rangesand times, most of these are extremely variable, i.e., age is a poorpredictor of performance. This was not so for visual perception.Although there are exceptions, for the most part visualperformance becomes progressively poorer with age, a processwhich accelerates somewhere in the fifth decade of life.

Some of these changes are attributable to optical andphysiological conditions in the aging eye, while others relate tochanges in neural processing of the image formed on the retina.There are other cognitive changes which are also central tounderstanding performance differences as drivers age. Both

CHANGES IN VISUAL PERCEPTION

Loss of Visual Acuity (static) - Fifteen to 25 percent of thepopulation 65 and older manifest visual acuities (Snellen) of less

(Marmor 1982). Peripheral vision is relatively unaffected,although a gradual narrowing of the visual field from 170 degreesto 140 degrees or less is attributable to anatomical changes (eyesbecome more sunk in the head). Static visual acuity amongdrivers is not highly associated with accident experience and isprobably not a very significant factor in discerning path guidancedevices and markings.

Light Losses and Scattering in Optic Train - There is someevidence (Ordy et al. 1982) that the scotopic (night) vision systemages faster than the photopic (daylight) system does. In addition,scatter and absorption by the stiff, yellowed, and possiblycateracted crystalline lens of the eye accounts for much less lighthitting the degraded retina. The pupil also becomes stiffer withage, and dilates less for a given amount of light impingement(which considering that the mechanism of pupillary size is in partdriven by the amount of light falling on the retina suggests actualphysical atrophy of the pupil--senile myosis). There is also morematter in suspension in the vitreous humor of the aged eye thanexists in the younger eye. The upshot is that only 30 percent ofthe light under daytime conditions that gets to the retina in a 20year old gets to the retina of a 60 year old. This becomes muchworse at night (as little as 1/16), and is exacerbated by thescattering effect of the optic train. Points of bright light aresurrounded by halos that effectively obscure less bright objectsin their near proximity. Blackwell and Blackwell (1971)estimated that, because of these changes, a given level of contrastof an object has to be increased by a factor of anywhere from 1.17to 2.51 for a 70 year old person to see it, as compared to a 30 yearold.

Glare Recovery - It is worth noting that a 55 year old personrequires more than 8 times the period of time to recover fromglare if dark adapted than a 16 year old does (Fox 1989). Anolder driver who does not use the strategy to look to the right andshield his or her macular vision from oncoming headlamp glare

3. HUMAN FACTORS

3 - 18

is literally driving blind for many seconds after exposure. As lived in before becoming "older drivers," and they have drivendescribed above, scatter in the optic train makes discerning anymarking or traffic control device difficult to impossible. The slowre-adaptation to mesopic levels of lighting is well-documented.

Figure/Ground Discrimination - Perceptual style changes withage, and many older drivers miss important cues, especially underhigher workloads (Fox 1989). This means drivers may miss asignificant guideline or marker under unfamiliar drivingconditions, because they fail to discriminate the object from itsbackground, either during the day or at night.

CHANGES IN COGNITIVE PERFORMANCE

Information Filtering Mechanisms - Older drivers reportedlyexperience problems in ignoring irrelevant information andcorrectly identifying meaningful cues (McPherson et al. 1988).Drivers may not be able to discriminate actual delineation orsignage from roadside advertising or faraway lights, for example.Work zone traffic control devices and markings that are meantto override the pre-work TCD's may be missed.

Forced Pacing under Highway Conditions - In tasks thatrequire fine control, steadiness, and rapid decisions, forced pacedtasks under stressful conditions may disrupt the performance ofolder drivers. They attempt to compensate for this by slowingdown. Older people drive better when they can control their ownpace (McPherson et al. 1988). To the traffic flow theorist, asizable proportion of older drivers in a traffic stream may resultin vehicles that lag behind and obstruct the flow.

Central vs. Peripheral Processes - Older driver safety problemsrelate to tasks that are heavily dependent on central processing.These tasks involve responses to traffic or to roadway conditions(emphasis added) (McPherson et al. 1988).

The Elderly Driver of the Past or Even of Today is Not theOlder Driver of the Future - The cohort of drivers who will be65 in the year 2000, which is less than five years from now, wereborn in the 1930's. Unlike the subjects of gerontology studiesdone just a few years ago featuring people who came of drivingage in the 1920's or even before, when far fewer people had carsand traffic was sparse, the old of tomorrow started driving in the1940's and after. They are and will be more affluent, bettereducated, in better health, resident in the same communities they

under modern conditions and the urban environment since theirteens. Most of them have had classes in driver education anddefensive driving. They will likely continue driving on a routinebasis until almost the end of their natural lives, which will behappening at an ever advancing age. The cognitive trends brieflydiscussed above are very variable in incidence and in their actualeffect on driving performance. The future older driver may wellexhibit much less decline in many of these performance areas inwhich central processes are dominant.

3.7.3 Driver Impairment

Drugs - Alcohol abuse in isolation and combination with otherdrugs, legal or otherwise, has a generally deleterious effect onperformance (Hulbert 1988; Smiley 1974). Performancedifferences are in greater variability for any given driver, and ingenerally lengthened reaction times and cognitive processingtimes. Paradoxically, some drug combinations can improve suchperformance on certain individuals at certain times. The onlydrug incidence which is sufficiently large to merit considerationin traffic flow theory is alcohol.

Although incidence of alcohol involvement in accidents has beenresearched for many years, and has been found to be substantial,very little is known about incidence and levels of impairment inthe driving population, other than it must also be substantial.Because these drivers are impaired, they are over-represented inaccidents. Price (1988) cites estimates that 92 percent of theadult population of the U.S. use alcohol, and perhaps 11 percentof the total adult population (20-70 years of age) have alcoholabuse problems. Of the 11 percent who are problem drinkers,seven percent are men, four percent women. The incidence ofproblem drinking drops with age, as might be expected. Effectson performance as a function of blood alcohol concentration(BAC) are well-summarized in Price, but are too voluminous tobe reproduced here. Price also summarizes effects of other drugssuch as cocaine, marijuana, etc. Excellent sources for moreinformation on alcohol and driving can be found in aTransportation Research Board Special Report (216).

Medical Conditions - Disabled people who drive represent asmall but growing portion of the population as technologyadvances in the field of adaptive equipment. Performance

g(s) �Ke �ts (1�TLs)

(1�TLs)(1�TNs)�R

3. HUMAN FACTORS

3 - 19

(3.12)

studies and insurance claim experience over the years with illnesses or conditions for which driving is contraindicated,(Koppa et al. 1980) suggest tha t such driver's performance they are probably not enough of these to account for them in anyis indistinguishable from the general driving population. traffic flow models.Although there are doubtless a number of people on the highways

3.8 Continuous Driver Performance

The previous sections of this chapter have sketched the relevant nothing can be more commonplace than steering a motor vehicle.discrete performance characteristics of the driver in a traffic Figure 3.5 illustrates the conceptual model first proposed bystream. Driving, however, is primarily a continuous dynamicprocess of managing the present heading and thus the future pathof the vehicle through the steering function. The first and secondderivatives of location on the roadway in time, velocity andacceleration, are also continuous control processes throughmodulated input using the accelerator (really the throttle) and thebrake controls.

3.8.1 Steering Performance

The driver is tightly coupled into the steering subsystem of thehuman-machine system we call the motor vehicle. It was onlyduring the years of World War II that researchers and engineersfirst began to model the human operator in a tracking situationby means of differential equations, i.e., a transfer function. Thefirst paper on record to explore the human transfer function wasby Tustin in 1944 (Garner 1967), and the subject was antiaircraftgun control. The human operator in such tracking situations canbe described in the same terms as if he or she is a linear feedbackcontrol system, even though the human operator is noisy, non-linear, and sometimes not even closed-loop.

3.8.1.1 Human Transfer Function for Steering

Steering can be classified as a special case of the general pursuittracking model, in which the two inputs to the driver (which aresomehow combined to produce the correction signal) are (1) thedesired path as perceived by the driver from cues provided by theroadway features, the streaming of the visual field, and higherorder information; and (2) the perceived present course of thevehicle as inferred from relationship of the hood to roadwayfeatures. The exact form of either of these two inputs are stillsubjects of investigation and some uncertainty, even though

Sheridan (1962). The human operator looks at both inputs, R(t)the desired input forcing function (the road and where it seemsto be taking the driver), and E(t) the system error function, thedifference between where the road is going and what C(t) thevehicle seems to be doing. The human operator can look ahead(lead), the prediction function, and also can correct for perceivederrors in the path. If the driver were trying to drive by viewingthe road through a hole in the floor of the car, then the predictionfunction would be lost, which is usually the state of affairs forservomechanisms. The two human functions of prediction andcompensation are combined to make a control input to the vehiclevia the steering wheel which (for power steering) is also a servoin its own right. The control output from this human-steeringprocess combination is fed back (by the perception of the pathof the vehicle) to close the loop. Mathematically, the setup inFigure 3.5 is expressed as follows, if the operator is reasonablylinear:

Sheridan (1962) reported some parameters for this Laplacetransform transfer function of the first order. K, the gain orsensitivity term, varies (at least) between +35 db to -12 db. Gain,how much response the human will make to a given input, is theparameter perhaps most easily varied, and tends to settle at somepoint comfortably short of instability (a phase margin of 60degrees or more). The exponential term e ranges from 0.12 to-ts

0.3 sec and is best interpreted as reaction time. This delay is thedominant limit to the human's ability to adapt to fast-changingconditions. The T factors are all time constants, which howevermay not stay constant at all. They must usually be empiricallyderived for a given control situation.

3. HUMAN FACTORS

3 - 20

Figure 3.5Pursuit Tracking Configuration (after Sheridan 1962).

Sheridan reported some experimental results which show TL

(lead) varying between 0 and 2, T (lag) from 0.0005 to 25, andl

T (neuromuscular lag) from 0 to 0.67. R, the remnant term, isNusually introduced to make up for nonlinearities between inputand output. Its value is whatever it takes to make output trackinput in a predictable manner. In various forms, and sometimeswith different and more parameters, Equation 3.10 expresses the Maneuver satorybasic approach to modeling the driver's steering behavior.Novice drivers tend to behave primarily in the compensatorytracking mode, in which they primarily attend to the difference,say, between the center of the hood and the edge line of thepavement, and attempt to keep that difference at some constantvisual angle. As they become more expert, they move more topursuit tracking as described above. There is also evidence thatthere are "precognitive" open-loop steering commands toparticular situations such as swinging into an accustomed parkingplace in a vehicle the driver is familiar with. McRuer and Klein(1975) classify maneuvers of interest to traffic flow modelers asis shown in Table 3.9.

In Table 3.9, the entries under driver control mode denote theorder in which the three kinds of tracking transition from one tothe other as the maneuver transpires. For example, for a turningmovement, the driver follows the dotted lines in an intersectionand aims for the appropriate lane in the crossroad in a pursuitmode, but then makes adjustments for lane position during thelatter portion of the maneuver in a compensatory mode. In anemergency, the driver "jerks" the wheel in a precognitive (open-loop) response, and then straightens out the vehicle in the newlane using compensatory tracking.

Table 3.9Maneuver Classification

Driver Control Mode

Compen- Pursuit Precognitive

Highway LaneRegulation

1

Precision CourseControl

2 1

Turning; RampEntry/Exit

2 1

Lane Change 2 1

Overtake/Pass 2 1

Evasive LaneChange

2 1

3.8.1.2 Performance Characteristics Based on Models

The amplitude of the output from this transfer function has beenfound to rapidly approach zero as the frequency of the forcingfunction becomes greater than 0.5 Hz (Knight 1987). The drivermakes smaller and smaller corrections as the highway or windgusts or other inputs start coming in more frequently than onecomplete cycle every two seconds.

gs �SRl (1�Fsu

2)Cr

1000

d �

V 2

257.9f

3. HUMAN FACTORS

3 - 21

(3.13)

(3.14)

The time lag between input and output also increases with Then a stage of steady-state curve driving follows, with the driverfrequency. Lags approach 100 msec at an input of 0.5 Hz and now making compensatory steering corrections. The steeringincrease almost twofold to 180 msec at frequencies of 2.4 Hz. wheel is then restored to straight-ahead in a period that covers theThe human tracking bandwidth is of the order of 1 to 2 Hz. endpoint of the curve. Road curvature (perceived) and vehicleDrivers can go to a precognitive rhythm for steering input to speed predetermines what the initial steering input will be, in thebetter this performance, if the input is very predictable, e.g., a following relationship:"slalom" course. Basic lane maintenance under very restrictiveconditions (driver was instructed to keep the left wheels of avehicle on a painted line rather than just lane keep) was studiedvery recently by Dulas (1994) as part of his investigation ofchanges in driving performance associated with in-vehicledisplays and input tasks. Speed was 57 km/h. Dulas foundaverage deviations of 15 cm, with a standard deviation of 3.2 cm. where,Using a tolerance estimation based on the nearly 1000observations of deviation, the 95th percentile deviation would be21 cm, the 99th would be 23 cm. Thus drivers can be expectedto weave back and forth in a lane in straightaway driving in anenvelope of +/- 23 cm or 46 cm across. Steering accuracy withdegrade and oscillation will be considerably more in curves. sincesuch driving is mixed mode, with rather large errors at thebeginning of the maneuver, with compensatory corrections towardthe end of the maneuver. Godthelp (1986) described this processas follows. The driver starts the maneuver with a lead term beforethe curve actually begins. This precognitive control actionfinishes shortly after the curve is entered.

C = roadway curvaturer

SR = steering ratioF = stability factors

l = wheelbaseu = speedg = steering wheel angle (radians)s

Godthelp found that the standard deviation of anticipatory steeringinputs is about 9 percent of steering wheel angle g . Since sharperscurves require more steering wheel input, inaccuracies will beproportionately greater, and will also induce more oscillation fromside to side in the curve during the compensatory phase of themaneuver.

3.9 Braking Performance

The steering performance of the driver is integrated with either or more wheels locking and consequent loss of control at speedsbraking or accelerator positioning in primary control input. higher than 32 km/h, unless the vehicle is equipped with antiskidHuman performance aspects of braking as a continuous control brakes (ABS, or Antilock Brake System). Such a model ofinput will be discussed in this section. After the perception- human braking performance is assumed in the time-hallowedresponse time lag has elapsed, the actual process of applying the AASHTO braking distance formula (AASHTO 1990):brakes to slow or stop the motor vehicle begins.

3.9.1 Open-Loop Braking Performance

The simplest type of braking performance is "jamming on thebrakes." The driver exerts as much force as he or she can muster,and thus approximates an instantaneous step input to the motorvehicle. Response of the vehicle to such an input is out of scopefor this chapter, but it can be remarked that it can result in one

where,d = braking distance - metersV = Initial speed - km/hf = Coefficient of friction, tires to pavement

surface, approximately equal to decelerationin g units

3. HUMAN FACTORS

3 - 22

Figure 3.6 shows what such a braking control input really looks Figure 3.7 shows a braking maneuver on a tangent with the samelike in terms of the deceleration profile. This maneuver was on driver and vehicle, this time on a wet surface. Note thea dry tangent section at 64 km/h, under "unplanned" conditions characteristic "lockup" footprint, with a steady-state deceleration(the driver does not know when or if the signal to brake will be after lockup of 0.4 g. given) with a full-size passenger vehicle not equipped withABS. Note the typical steep rise in deceleration to a peak of over From the standpoint of modeling driver input to the vehicle, the0.9 g, then steady state at approximately 0.7 g for a brakes locked open-loop approximation is a step input to maximum brakingstop. The distance data is also on this plot: the braking distance effort, with the driver exhibiting a simple to complex PRT delayon this run was 23 m feet. Note also that the suspension bounce prior to the step. A similar delay term would be introduced priorproduces a characteristic oscillation after the point at which the to release of the brake pedal, thus braking under stop-and-govehicle is completely stopped, just a little less than five seconds conditions would be a sawtooth. into the run.

Figure 3.6Typical Deceleration Profile for a Driver without

Antiskid Braking System on a Dry Surface.

3. HUMAN FACTORS

3 - 23

Figure 3.7Typical Deceleration Profile for a Driver without

Antiskid Braking System on a Wet Surface.

3.9.2 Closed-Loop Braking Performance

Recent research in which the writer has been involved providesome controlled braking performance data of direct applicationto performance modeling (Fambro et al. 1994). "Steady state"approximations or fits to these data show wide variations amongdrivers, ranging from -0.46 g to -0.70 g.

Table 3.10 provides some steady-state derivations from empiricaldata collected by Fambro et al. (1994). These were all responsesto an unexpected obstacle or object encountered on a closedcourse, in the driver's own (but instrumented) car.

Table 3.11 provides the same derivations from data collected ondrivers in their own vehicle in which the braking maneuver wasanticipated; the driver knows that he or she would be braking, butduring the run were unsure when the signal (a red light inside thecar) would come.

The ratio of unexpected to expected closed-loop braking effortwas estimated by Fambro et al. to be about 1.22 under the same

Table 3.10Percentile Estimates of Steady State UnexpectedDeceleration

Mean -0.55g

Standard Deviation 0.07

75th Percentile -0.43

90th -0.37

95th -0.32

99th -0.24

pavement conditions. Pavement friction (short of ice) playedvery little part in driver's setting of these effort levels. About0.05 to 0.10 g difference between wet pavement and drypavement steady-state g was found.

3. HUMAN FACTORS

3 - 24

Table 3.11 to come to rest. Driver input to such a planned braking situationPercentile Estimates of Steady-State ExpectedDeceleration

Mean -0.45g

Standard Deviation 0.09 maximum "comfortable" braking deceleration is generally

75th Percentile -0.36

90th -0.31

95th -0.27

99th -0.21

3.9.3 Less-Than-MaximumBraking Performance

The flow theorist may require an estimate of “comfortable”braking performance, in which the driver makes a stop forintersections or traffic control devices which are discernedconsiderably in advance of the location at which the vehicle is

approximates a "ramp" (straight line increasing with time fromzero) function with the slope determined by the distance to thedesired stop location or steady-state speed in the case of aplatoon being overtaken. The driver squeezes on pedal pressureto the brakes until a desired deceleration is obtained. The

accepted to be in the neighborhood of -0.30 g, or around 3m/sec (ITE 1992). 2

The AASHTO Green Book (AASHTO 1990) provides a graphicfor speed changes in vehicles, in response to approaching anintersection. When a linear computation of decelerations fromthis graphic is made, these data suggest decelerations in theneighborhood of -2 to -2.6 m/sec or -0.20 to -0.27 g. More2

recent research by Chang et al. (1985) found values in responseto traffic signals approaching -0.39 g, and Wortman andMatthias (1983) observed a range of -0.22 to -0.43 g, with amean level of -0.36 g. Hence controlled braking performancethat yields a g force of about -0.2 g would be a reasonable lowerlevel for a modeler, i.e., almost any driver could be expected tochange the velocity of a passenger car by at least that amount,but a more average or "typical" level would be around -0.35 g.

3.10 Speed and Acceleration Performance

The third component to the primary control input of the driver at the moment, etc. Drivers in heavy traffic use relativeto the vehicle is that of manual (as opposed to cruise control) perceived position with respect to other vehicles in the streamcontrol of vehicle velocity and changes in velocity by means of as a primary tracking cue (Triggs, 1988). A recent studythe accelerator or other device to control engine RPM. (Godthelp and Shumann 1994) found errors between speed

3.10.1 Steady-State Traffic Speed Control

The driver's primary task under steady-state traffic conditionsis to perform a tracking task with the speedometer as the display,and the accelerator position as the control input. Driverresponse to the error between the present indicated speed andthe desired speed (the control signal) is to change the pedalposition in the direction opposite to the trend in the errorindication. How much of such an error must be present dependsupon a host of factors: workload, relationship of desired speedto posted speed, location and design of the speedometer, and The performance characteristics of the vehicle driver are thepersonal considerations affecting the performance of the driver limiting constraints on how fast the driver can accelerate the

desired and maintained to vary from -0.3 to -0.8 m/sec in a lanechange maneuver; drivers tended to lose velocity when theymade such a maneuver. Under steady-stage conditions in a trafficstream, the range of speed error might be estimated to be nomore than +/- 1.5 m/sec (Evans and Rothery 1973), basicallymodeled by a sinusoid. The growing prevalence of cruisecontrols undoubtedly will reduce the amplitude of this speederror pattern in a traffic stream by half or more.

3.10.2 Acceleration Control

aGV � aLV �

Gg

100

3. HUMAN FACTORS

3 - 25

(3.15)

vehicle. The actual acceleration rates, particularly in a traffic 1992). If the driver removes his or her foot from the acceleratorstream as opposed to a standing start, are typically much lower pedal (or equivalent control input) drag and rolling resistancethan the performance capabilities of the vehicle, particularly a produce deceleration at about the same level as "unhurried"passenger car. A nominal range for "comfortable" acceleration acceleration, approximately 1 m/sec at speeds of 100 km/h orat speeds of 48 km/h and above is 0.6 m/sec to 0.7 m/sec higher. In contrast to operation of a passenger car or light truck,2 2

(AASHTO 1990). Another source places the nominal heavy truck driving is much more limited by the performanceacceleration rate drivers tend to use under "unhurried" capabilities of the vehicle. The best source for such informationcircumstances at approximately 65 percent of maximumacceleration for the vehicle, somewhere around 1 m/sec (ITE2

2

is the Traffic Engineering Handbook (ITE 1992).

3.11 Specific Maneuvers at the Guidance Level

The discussion above has briefly outlined most of the morefundamental aspects of driver performance relevant to modelingthe individual driver-vehicle human-machine system in a trafficstream. A few additional topics will now be offered to furtherrefine this picture of the driver as an active controller at theguidance level of operation in traffic.

3.11.1 Overtaking and Passingin the Traffic Stream

3.11.1.1 Overtaking and Passing Vehicles(4-Lane or 1-Way)

Drivers overtake and pass at accelerations in the sub-maximalrange in most situations. Acceleration to pass another vehicle(passenger cars) is about 1 m/sec at highway speeds (ITE2

1992). The same source provides an approximate equation foracceleration on a grade:

where,a = max acceleration rate on gradeGV

a = max acceleration rate on levelLV

G = gradient (5/8)g = acceleration of gravity (9.8 m/sec )2

The maximum acceleration capabilities of passenger vehiclesrange from almost 3 m/sec from standing to less than 2 m/sec2 2

from 0 to highway speed.

Acceleration is still less when the maneuver begins at higherspeeds, as low as 1 m/sec on some small subcompacts. In2

Equation 3.15, overtaking acceleration should be taken as 65percent of maximum (ITE 1992). Large trucks or tractor-trailercombinations have maximum acceleration capabilities on a levelroadway of no more than 0.4 m/sec at a standing start, and2

decrease to 0.1 m/sec at speeds of 100 km/h. Truck drivers2

"floorboard" in passing maneuvers under these circumstances,and maximum vehicle performance is also typical driver input.

3.11.1.2 Overtaking and Passing Vehicles (Opposing Traffic)

The current AASHTO Policy on Geometric Design (AASHTO1990) provides for an acceleration rate of 0.63 m/sec for an2

initial 56 km/h, 0.64 m/sec for 70 km/h, and 0.66 m/sec for2 2

speeds of 100 km/h. Based upon the above considerations, thesedesign guidelines appear very conservative, and the theorist maywish to use the higher numbers in Section 3.11.1.1 in asensitivity analysis.

3. HUMAN FACTORS

3 - 26

3.12 Gap Acceptance and Merging

3.12.1 Gap Acceptance

The driver entering or crossing a traffic stream must evaluate Table 3.12 provides very recent design data on these situationsthe space between a potentially conflicting vehicle and himself from the Highway Capacity Manual (TRB 1985). The range ofor herself and make a decision whether to cross or enter or not. gap times under the various scenarios presented in Table 3.12The time between the arrival of successive vehicles at a point is from a minimum of 4 sec to 8.5 sec. In a stream traveling atis the time gap, and the critical time gap is the least amount of 50 km/h (14 m/sec) the gap distance thus ranges from 56 to 119successive vehicle arrival time in which a driver will attempt a m; at 90 km/h (25 m/sec) the corresponding distances are 100merging or crossing maneuver. There are five different gap to 213 m. acceptance situations. These are:

(1) Left turn across opposing traffic, no traffic control(2) Left turn across opposing traffic, with traffic control

(permissive green)(3) Left turn onto two-way facility from stop or yield

controlled intersection(4) Crossing two-way facility from stop or yield controlled

intersection(5) Turning right onto two-way facility from stop or yield

controlled intersection

3.12.2 Merging

In merging into traffic on an acceleration ramp on a freeway orsimilar facility, the Situation (5) data for a four lane facility at90 km/h with a one second allowance for the ramp provides abaseline estimate of gap acceptance: 4.5 seconds. Theoreticallyas short a gap as three car lengths (14 meters) can be acceptedif vehicles are at or about the same speed, as they would be inmerging from one lane to another. This is the minimum,however, and at least twice that gap length should be used as anominal value for such lane merging maneuvers.

3.13 Stopping Sight Distance

The minimum sight distance on a roadway should be sufficient empirically derived estimates now available in Fambro et al.to enable a vehicle traveling at or near the design speed to stop (1994) for both these parts of the SSD equation are expressedbefore reaching a "stationary object" in its path, according to the in percentile levels of drivers who could be expected to (1)AASHTO Policy on Geometric Design (AASHTO 1990). Itgoes on to say that sight distance should be at least that requiredfor a "below-average" driver or vehicle to stop in this distance.

Previous sections in this chapter on perception-response time(Section 3.3.1) and braking performance (Section 3.2) providethe raw materials for estimating stopping sight distance. Thetime-honored estimates used in the AASHTO Green Book(AASHTO 1990) and therefore many other engineeringresources give a flat 2.5 sec for PRT, and then the lineardeceleration equation (Equation 3.13) as an additive model.This approach generates standard tables that are used to estimatestopping sight distance (SSD) as a function of coefficients offriction and initial speed at inception of the maneuver. The

respond and (2) brake in the respective distance or shorter.Since PRT and braking distance that a driver may achieve in agiven vehicle are not highly correlated, i.e., drivers that may bevery fast to initiate braking may be very conservative in theactual braking maneuver, or may be strong brakers. PRT doesnot predict braking performance, in other words.

Very often, the engineer will use "worst case" considerations ina design analysis situation. What is the "reasonable" worst casefor achieving the AASHTO "below average" driver and vehicle?Clearly, a 99th percentile PRT and a 99th percentile brakingdistance gives an overly conservative 99.99 combined percentile

3. HUMAN FACTORS

3 - 27

Table 3.12Critical Gap Values for Unsignalized Intersections

Maneuver Control

Average Speed of Traffic

50 km/h 90 km/h

Number of Traffic Lanes, Major Roadway

2 4 2 4

1 None 5.0 5.5 5.5 6.0

2 Permissive 5.0 5.5 5.5 6.0Green1

3 Stop 6.5 7.0 8.0 8.5

3 Yield 6.0 6.5 7.0 7.5

4 Stop 6.0 6.5 7.5 8.0

4 Yield 5.5 6.0 6.5 7.0

5 Stop 5.5 5.5 6.5 6.52

5 Yield 5.0 5.0 5.5 5.52

During Green Interval1

If curve radius >15 m or turn angle <60�� subtract 0.5 seconds.2

If acceleration lane provided, subtract 1.0 seconds.

All times are given in seconds.

All Maneuvers: if population >250,000, subtract 0.5 secondsif restricted sight distance, add 1.0 secondsMaximum subtraction is 1.0 secondsMaximum critical gap �� 8.5 seconds

level--everybody will have an SSD equal to or shorter than thissomewhat absurd combination. The combination of 90thpercentile level of performance for each segment yields acombined percentile estimate of 99 percent (i.e., 0.10 of eachdistribution is outside the envelope, and their product is 0.01,therefore 1.00 - 0.01 = 0.99). A realistic worst case ( 99thpercentile) combination to give SSD would, from previoussections of this chapter be:

PRT: 1.57 sec (Table 3.4)

Braking deceleration: -0.37 g (Section 3.10.2) would be:

For example, on a dry level roadway, using Equation 3.12, at avelocity of 88 km/h, the SSD components would be:

PRT: 1.57 x 24.44 = 38.4 m

Braking Distance: 82.6 m

SSD: 38 + 83 = 121 m

For comparison, the standard AASHTO SSD for a dry levelroadway, using a nominal 0.65 for f, the coefficient of friction,

3. HUMAN FACTORS

3 - 28

PRT: 2.50 x 24.44 = 61.1 m These two estimates are comparable, but the first estimate has

Braking Distance: 47.3 m combinations of percentiles (for example, 75th percentile

SSD: 61 + 47 = 108 m percentile). It is always possible, of course, to assume different

an empirical basis for it. The analyst can assume other

performance in combination yields an estimate of the 94th

levels of percentile representation for a hypothetical driver, e.g.,50th percentile PRT with 95th percentile braking performance.

3.14 Intersection Sight Distance

The AASHTO Policy on Geometric Design (AASHTO 1990)identifies four differences cases for intersection sight distanceconsiderations. From the viewpoint of traffic flow theory, thequestion may be posed, "How long is a driver going to linger atan intersection before he or she begins to move?" Only the firstthree cases will be discussed here, since signalized intersections(Case IV) have been discussed in Section 3.5.1.

3.14.1 Case I: No Traffic Control

The driver initiates either acceleration or deceleration basedupon his or her perceived gap in intersecting traffic flow. Theprinciples given in Section 3.13 apply here. PRT for thissituation should be the same as for conditions of no surpriseoutlined in Section 3.3.1. AASHTO (1990) gives an allowanceof three seconds for PRT, which appears to be very conservativeunder these circumstances.

3.14.2 Case II: Yield Control forSecondary Roadway

This is a complex situation. McGee et al. (1983) could not findreliable data to estimate the PRT. A later, follow-up study byHostetter et al. (1986) considered the PRT to stretch from thetime that the YIELD sign first could be recognized as such tothe time that the driver either began a deceleration maneuver orspeeded up to clear the intersection in advance of cross traffic.But decelerations often started 300 m or more from theintersection, a clear response to the sign and not to the trafficahead. PRTs were thus in the range of 20 to over 30 sec. withmuch variability and reflect driving style rather thanpsychophysical performance.

3.14.3 Case III: Stop Control onSecondary Roadway

Hostetter et al. (1986) note that "for a large percentage of trialsat intersections with reasonable sign distance triangles, driverscompleted monitoring of the crossing roadway before comingto a stop." Their solution to this dilemma was to include threemeasures of PRT. They start at different points but terminatewith the initiation of an accelerator input. One of the PRT'sstarts with the vehicle at rest. The second begins with the firsthead movement at the stop. The third begins with the last headmovement in the opposite direction of the intended turn ortoward the shorter sight distance leg (for a through maneuver).None of the three takes into account any processing the drivermight be doing prior to the stop at the intersection.

Their findings were as follows (Table 3.13):

Table 3.13PRTs at Intersections

4-way T-Intersection

Mean 85th Mean 85th

PRT 1 2.2 sec 2.7 2.8 3.1

PRT 2 1.8 2.6 1.9 2.8

PRT 3 1.6 2.5 1.8 2.5

Thus a conservative estimate of PRT, i.e., time lag at anintersection before initiating a maneuver, would be somewhatin excess of three seconds for most drivers.

3. HUMAN FACTORS

3 - 29

3.15 Other Driver Performance Characteristics

3.15.1 Speed Limit Changes 3.15.3 Real-Time Driver Information Input

Gstalter and Hoyos (1988) point out the well-known With the advent of Intelligent Transportation Systems (ITS),phenomenon that drivers tend to adapt to sustained speed over driver performance changes associated with increaseda period of time, such that the perceived velocity (not looking information processing work load becomes a real possibility.at the speedometer) lessens. In one study cited by these authors, ITS may place message screens, collision avoidance displaysdrivers drove 32 km at speeds of 112 km/h. Subjects were then and much more in the vehicle of the future. Preliminaryinstructed to drop to 64 km/h. The average speed error turned studies of the effects of using such technology in a traffic streamout to be more than 20 percent higher than the requested speed.A similar effect undoubtedly occurs when posted speed limitschange along a corridor. In the studies cited, drivers were awareof the "speed creep" and attempted (they said) to accommodatefor it. When drivers go from a lower speed to a higher one, theyalso adapt, such that the higher speed seems higher than in factit is, hence errors of 10 to 20 percent slower than commandedspeed occur. It takes several minutes to re-adapt. Hence speedadjustments on a corridor should not be modeled by a simplestep function, but rather resemble an over damped first-orderresponse with a time constant of two minutes or more.

3.15.2 Distractors On/Near Roadway

One of the problems that militates against smooth traffic flowon congested facilities is the "rubber neck" problem. Driverspassing by accident scenes, unusual businesses or activities onthe road side, construction or maintenance work, or otheroccurrences irrelevant to the driving task tend to shift sufficientattention to degrade their driving performance. In PositiveGuidance terms (Lunenfeld and Alexander 1990) such asituation on or near the roadway is a temporary violation ofexpectancy. How to model the driver response to suchdistractions? In the absence of specific driver performance dataon distractors, the individual driver response could be estimatedby injecting a sudden accelerator release with consequentdeceleration from speed discussed in Section 3.11.2. Thisresponse begins as the distraction comes within a cone of 30degrees centered around the straight-ahead direction on atangent, and the outer delineation of the curve on a horizontalcurve. A possible increase in the amplitude of lane excursionscould also occur, similar to the task-loaded condition in theDulas (1994) study discussed in Section 3.9.1.2.

are just now appearing in the literature. There is clearly muchmore to come. For a review of some of the human factorsimplications of ITS, see Hancock and Pansuraman (1992) andany recent publications by Peter Hancock of The University ofMinnesota.

Drivers as human beings have a very finite attentional resourcecapacity, as summarized in Dulas (1994). Resources can beallocated to additional information processing tasks only at thecost of decreasing the efficiency and accuracy of those tasks.When the competing tasks use the same sensory modality andsimilar resources in the brain, increases in errors becomesdramatic. To the extent that driving is primarily a psychomotortask at the skill-based level of behavior, it is relatively immuneto higher-level information processing, if visual perception isnot a dominant factor.

But as task complexity increases, say, under highly congestedurban freeway conditions, any additional task becomes verydisruptive of performance. This is especially true of olderdrivers. Even the use of cellular telephones in traffic has beenfound to be a potent disrupter of driving performance (McKnightand McKnight 1993). A study described by Dulas (1994) foundthat drivers using a touch screen CRT at speeds of 64 km tendedto increase lane deviations such that the probability of laneexcursion was 0.15. Early and very preliminary studies indicatethat close attention to established human engineering principlesfor information display selection and design should result in real-time information systems that do not adversely affect driverperformance.

3. HUMAN FACTORS

3 - 30

ReferencesAmerican Association of Highway and Transportation

Officials (1990). A Policy on Geometric Design ofHighways and Streets. Washington, DC. Journal of Experimental Psychology, 47, pp. 381-391.

Ang, A. H-S., and W. H. Tang (1975). Probability Concepts Fox, M. (1989). Elderly Driver's Perceptions of Theirin Engineering Planning and Design Vol. 1: Basic Driving Abilities Compared to Their Functional VisualPrinciples. New York: John Wiley and Sons. Perception Skills and Their Actual Driving Performance.

Berman, A. H. (1994). A Study of Factors Affecting FootMovement Time in a Braking Maneuver. Unpublished MSThesis, Texas A&M University.

Blackwell, O. and H. Blackwell (1971). Visual PerformanceData for 156 Normal Observers of Various Ages. Journalof Illumination Engineering Society 1, pp. 3-13.

Boff, K. R. and J. E. Lincoln (1988). Engineering Data Compendium. USAF, H. G. Armstrong Medical Research

Laboratory, Wright-Patterson AFB, Ohio, (Section 8.1).Brackett, R. Q. and R. J. Koppa (1988). Preliminary Study

of Brake Pedal Location Accuracy. Proceedings of theHuman Factors and Ergonomics Society 32nd Meeting.

Chang, M-S, C. J. Messer, and A. J. Santiago (1985). TimingTraffic Signal Change Intervals Based on DriverBehavior. Transportation Research Record 1027,Transportation Research Board, National ResearchCouncil, Washington, DC.

Drury, C. G. (1975). Application of Fitt's Law to Foot Pedal Design. Human Factors 12, pp. 559-561.Dudek, C. L. (1990). Guidelines on the Use of Changeable

Message Signs. FHWA-TS-90-043, Federal HighwayAdministration, Washington, DC.

Dulas, R. J. (1994). Lane Deviation as a Result of VisualWorkload and Manual Data Entry. Unpublished doctoraldissertation, Texas A&M University.

Evans, L. and R. Rothery (1973). ExperimentalMeasurements of Perceptual Thresholds in CarFollowing. Highway Research Record 454, TransportationResearch Board, Washington, DC.

Fambro, D. et al. (1994). Determination of Factors AffectingStopping Sight Distance. Working Paper I: BrakingStudies (DRAFT) National Cooperative Highway ResearchProgram, Project 3-42.

Federal Highway Administration (1983). Traffic Control Device Handbook.Washington, DC.

Fitts, P. M. (1954). The Information Capacity of the HumanMotor System in Controlling the Amplitude of Movement.

Taira, E. (Ed.). Assessing the Driving Ability of theElderly New York: Hayworth Press.

Garner, K.C. (1967). Evaluation of Human OperatorCoupled Dynamic Systems. In Singleton, W. T., Easterby,R. S., and Whitfield, D. C. (Eds.). The Human Operatorin Complex Systems, London: Taylor and Francis.

Godthelp, H. (1986). Vehicle Control During Curve Driving.Human Factors, 28,2, pp. 211-221.

Godthelp, H. and J. Shumann (1994). Use of IntelligentAccelerator as an Element of a Driver Support System.Abstracted in Driver Performance Data Book Update:Older Drivers and IVHS Circular 419, TransportationResearch Board, National Research Council, Washington,DC.

Greene, F. A. (1994). A Study of Field and LaboratoryLegibility Distances for Warning Symbol Signs.Unpublished doctoral dissertation, Texas A&M University.

Gstalter, H. and C. G. Hoyos (1988). Human Factors Aspectsof Road Traffic Safety. In Peters, G.A. and Peters, B.J.,Automotive Engineering and Litigation Volume 2, NewYork: Garland Law Publishing Co.

Hancock, P. A. and R. Parasuraman (1992). Human Factorsand Safety in the Design of Intelligent Vehicle - HighwaySystems (IVHS).

Hoffman, E. R. (1991). Accelerator to Brake MovementTimes and A Comparison of Hand and Foot MovementTimes. Ergonomics 34, pp. 277-287 and pp. 397-406.

Hooper, K. G. and H. W. McGee (1983). Driver Perception-Reaction Time: Are Revisions to Current Specificationsin Order? Transportation Research Record 904,Transportation Research Board, National ResearchCouncil, Washington, DC, pp. 21-30.

Hostetter, R., H. McGee, K. Crowley, E. Seguin, and G. Dauber,(1986). Improved Perception-Reaction Time Informationfor Intersection Sight Distance. FHWA/RD-87/015,Federal Highway Administration, Washington, DC.

3. HUMAN FACTORS

3 - 31

Hulbert, S. (1988). The Driving Task. In Peters, G.A. and McKnight, A. J. and A. S. McKnight (1993). The EffectPeters, B.J., Automotive Engineering and LitigationVolume 2, New York: Garland Law Publishing Co.

Institute of Transportation Engineers (1992). Traffic McPherson, K., J. Michael, A. Ostrow, and P. Shaffron Engineering Handbook, Washington, DC. (1988). Physical Fitness and the Aging Driver - Phase I.

IVHS America Publication B-571, Society of Automotive AAA Foundation for Traffic Safety, Washington, DC.Engineers, (1992). Intelligent Vehicle-Highway Systems McRuer, D. T. and R. H. Klein (1975). Automobile(IVHS) Safety and Human Factors Considerations. Controllability - Driver/Vehicle Response for SteeringWarrendale, PA.

Jacobs, R. J., A. W. Johnston, and B. L. Cole, (1975). The HS-801-407, National Highway Traffic Safety Visibility of Alphabetic and Symbolic Traffic Signs.

Australian Board Research, 5, pp. 68-86. Moir, A. and D. Jessel (1991). Brain Sex. New York:Johansson, G. and K. Rumar (1971). Driver's Brake Reaction

Times. Human Factors, 12 (1), pp. 23-27.Knight, J. L. (1987). Manual Control of Tracking in

Salvendy, G. (ed) Handbook of Human Factors. New Litigation,Vol. 2, New York: Garland Law Publishing Co.York: John Wiley and Sons. NHTSA Driver Performance Data Book (1987).

Koppa, R. J. (1990). State of the Art in Automotive Adaptive Equipment. Human Factors, 32 (4), pp. 439-455.

Koppa, R. J., M. McDermott, et al., (1980). Human Factors Analysis of Automotive Adaptive Equipment for Disabled

Drivers. Final Report DOT-HS-8-02049, NationalHighway Traffic Safety Administration, Washington, DC.

Kroemer, K. H. E., H. B. Kroemer, and K. E. Kroemer-Ebert (1994). Ergonomics: How to Design for Ease and

Efficiency Englewood Cliffs, NJ: Prentice-Hall.Lerner, Neil, et al. (1995). Literature Review: Older Driver

Perception-Reaction Time for Intersection Sight Distanceand Object Detection, Comsis Corp., Contract DTFH 61-90-00038 Federal Highway Administration, Washington,DC, Final Report.

Lunenfeld, H. and G. J. Alexander (1990). A User's Guide to Positive Guidance (3rd Edition) FHWA SA-90-017,

Federal Highway Administration, Washington, DC.Marmor, M. (1982). Aging and the Retina. In R. Sekuler,

D. Kline, and K. Dismukes (eds.). Aging and HumanVisual Function, New York: Alan R. Liss.

McGee, H. W. (1989). Reevaluation of Usefulness andApplication of Decision Sight Distance. TransportationResearch Record 1208, TRB, NRC, Washington, DC.

McGee, H., K. Hooper, W. Hughes, and W. Benson (1983). Highway Design and Operational Standards Affected byDriver Characteristics. FHWA RD 78-78, FederalHighway Administration, Washington, DC.

of Cellular Phone Use Upon Driver Attention. AccidentAnalysis and Prevention 25 (3), pp. 259-265.

Control. Systems Technology Inc. TR-1040-1-I. DOT-

Administration, Washington, DC.

Carol Publishing Co.Mortimer, R. G. (1988). Rear End Crashes. Chapter 9 of

Peters, G.A. and Peters, B.J., Automotive Engineering and

R. L. Henderson (ed.) DOT HS 807 121, National HighwayTraffic Safety Administration, Washington, DC.

NHTSA Driver Performance Data Book (1994).Transportation Reserach Circular 419, TransportationResearch Board, National Research Council, Washington,DC.

Neuman, T. R. (1989). New Approach to Design forStopping Sight Distance. Transportation Research Record,1208, Transportation Research Board, National ResearchCouncil, Washington, DC.

O'Leary, A. and R. Atkins (1993). Transportation Needs ofthe Older Driver. Virginia Transportation ResearchCouncil MS-5232 (FHWA/VA-93-R14).

Odeh, R. E. (1980). Tables for Normal Tolerance Limits,Sampling Plans, and Screening. New York: arcel Dekker,Inc.

Ordy, J. , K. Brizzee, and H. Johnson (1982 ). Cellular Alterations in Visual Pathways and the Limbic System:

Implications for Vision and Short Term Memory. ibid.Picha, D. (1992). Determination of Driver Capability

in the Detection and Recognition of an Object. InKahl, K. B. Investigation of Appropriate Objects for SafeStopping Distance. Unpublished Master's Thesis, TexasA&M University.

Price, D. (1988). Effects of Alcohol and Drugs. In Peters, G. A. and Peters, B. J., Automotive Engineering

and Litigation Volume 2, New York: Garland LawPublishing Co.

3. HUMAN FACTORS

3 - 32

Reason, J. (1990). Human Error. New York: Cambridge University Press. (1988). Transportation in an Aging Society. SpecialSchiff, W. (1980). Perception: An Applied Approach.

Boston: Houghton Mifflin.Sekuler, R. and R. Blake (1990). Perception. 2nd Ed. New

York: McGraw-Hill.Sheridan, T. B. (1962). The Human Operator in Control

Instrumentation. In MacMillan, R. H., Higgins, T. J., andNaslin, P. (Eds.)

Simmala, H. (1981). Driver/Vehicle Steering ResponseLatencies. Human Factors 23, pp. 683-692.

Smiley, A. M. (1974). The Combined Effect of Alcoholand Common Psychoactive Drugs: Field Studies with anInstrumented Automobile. Technical Bulletin ST-738,National Aeronautical Establishment, Ottawa, Canada.

Taoka, G. T. (1989). Brake Reaction Times of Unalerted Drivers. ITE Journal, Washington, DC.Transportation Research Board (1985). Highway Capacity

Manual, Special Report 209, Washington, DC.Transportation Research Board (1987). Zero Alcohol and

Other Options - Limits for Truck and Bus Drivers, SpecialReport 216, Transportation Research Board, NationalResearch Council, Washington, DC.

Transportation Research Board, National Research Council

Report 218, Washington, DC.Transportation Research Board, National Research Council

(1993). Human Factors Research in Highway Safety,TRB Circular 414, Washington, DC.

Triggs, T. J. (1988). Speed Estimation, Chapter 18, Vol. II,Peters, G. A. and Peters, B. Automotive Engineering andLitigation, op. cit.

Weir, David, (1976). Personal communication at a meeting,CA.

Wierwille, W. W., J. G. Casali, and B. S. Repa (1983).Driver Steering Reaction Time to Abrupt-onsetCrosswinds as Measured in a Moving Base DrivingSimulator. Human Factors 25, pp. 103-116.

Wortman, R. H. and J. S. Matthias (1983). Evaluation ofDriver Behavior at Signalized Intersections.Transportation Research Record 904, TRB, NRC,Washington, DC.

Senior Lecturer, Civil Engineering Department, The University of Texas, ECJ Building 6.204, Austin, TX 6

78712

CAR FOLLOWING MODELS

BY RICHARD W. ROTHERY6

T

xf(t)

x�(t)

xf(t)

�x�(t)

�xf(t)

x�(t)

xf(t)

xi(t)


� = Numerical coefficientsa = Generalized sensitivity coefficient

�,ma (t) = Instantaneous acceleration of a followingf

vehicle at time ta (t) = Instantaneous acceleration of a lead

�

vehicle at time t� = Numerical coefficient C = Single lane capacity (vehicle/hour)� = Rescaled time (in units of response time,

T) following vehicle� = Short finite time period u (t) = Velocity profile of the lead vehicle of aF = Amplitude factor� = Numerical coefficient k = Traffic stream concentration in vehicles

per kilometer � = Frequency of a monochromatic speedk = Jam concentrationjk = Concentration at maximum flowmk = Concentration where vehicle to vehiclef

interactions begin = Instantaneous speed of a lead vehicle atk = Normalized concentration time tnL = Effective vehicle lengthL = Inverse Laplace transform-1

� = Proportionality factor� = Sensitivity coefficient, i = 1,2,3,...iln(x) = Natural logarithm of xq = Flow in vehicles per hourq = Normalized flown<S> = Average spacing rear bumper to rear

bumper = Instantaneous position of the followingS = Initial vehicle spacing vehicle at time tiS = Final vehicle spacing = Instantaneous position of the ith vehicle atfS = Vehicle spacing for stopped traffic time toS(t) = Inter-vehicle spacing z(t) = Position in a moving coordinate system�S = Inter-vehicle spacing change

= Average response timeT = Propagation time for a disturbanceot = Time

t = Collision timecT = Reaction timeU = Speed of a lead vehicle

�

U = Speed of a following vehiclefU = Final vehicle speedfU = Free mean speed, speed of traffic near zerof

concentrationU = Initial vehicle speediU = Relative speed between a lead andrel

�

platoonV = Speed V = Final vehicle speedf

oscillation= Instantaneous acceleration of a following

vehicle at time t

= Instantaneous speed of a following vehicleat time t

= Instantaneous speed of a lead vehicle attime t

= Instantaneous speed of a following vehicleat time t

= Instantaneous position of a lead vehicle attime t

< x > = Average of a variable x

� = Frequency factor

S � ��V��V 2

� � 0.5(a �1f �a �1

�)

� � �

(4.2)

(4.3)

4.CAR FOLLOWING MODELS

It has been estimated that mankind currently devotes over 10 roadways. The speed-spacing relations that were obtained frommillion man-years each year to driving the automobile, which on these studies can be represented by the following equation:demand provides a mobility unequaled by any other mode oftransportation. And yet, even with the increased interest intraffic research, we understand relatively little of what isinvolved in the "driving task". Driving, apart from walking,talking, and eating, is the most widely executed skill in the worldtoday and possibly the most challenging. where the numerical values for the coefficients, �, �, and � take

Cumming (1963) categorized the various subtasks that are are given below:involved in the overall driving task and paralleled the driver'srole as an information processor (see Chapter 3). This chapterfocuses on one of these subtasks, the task of one vehiclefollowing another on a single lane of roadway (car following).This particular driving subtask is of interest because it isrelatively simple compared to other driving tasks, has beensuccessfully described by mathematical models, and is animportant facet of driving. Thus, understanding car followingcontributes significantly to an understanding of traffic flow. Carfollowing is a relatively simple task compared to the totality oftasks required for vehicle control. However, it is a task that iscommonly practiced on dual or multiple lane roadways whenpassing becomes difficult or when traffic is restrained to a singlelane. Car following is a task that has been of direct or indirectinterest since the early development of the automobile.

One aspect of interest in car following is the average spacing, S,that one vehicle would follow another at a given speed, V. Theinterest in such speed-spacing relations is related to the fact thatnearly all capacity estimates of a single lane of roadway werebased on the equation:

C = (1000) V/S (4.1) permeated safety organizations can be formed. In general, the

where physical length of vehicles; the human-factor element ofC = Capacity of a single lane

(vehicles/hour) V = Speed (km/hour)S = Average spacing rear bumper to rear

bumper in meters

The first Highway Capacity Manual (1950) lists 23observational studies performed between 1924 and 1941 thatwere directed at identifying an operative speed-spacing relationso that capacity estimates could be established for single lanes of

on various values. Physical interpretations of these coefficients

� = the effective vehicle length, L� = the reaction time, T� = the reciprocal of twice the maximum average

deceleration of a following vehicle

In this case, the additional term, � V , can provide sufficient2

spacing so that if a lead vehicle comes to a full stopinstantaneously, the following vehicle has sufficient spacing tocome to a complete stop without collision. A typical valueempirically derived for � would be � 0.023 seconds /ft . A less2

conservative interpretation for the non-linear term would be:

where a and a are the average maximum decelerations of theƒ �

following and lead vehicles, respectively. These terms attemptto allow for differences in braking performances betweenvehicles whether real or perceived (Harris 1964).

For � = 0, many of the so-called "good driving" rules that have

speed-spacing Equation 4.2 attempts to take into account the

perception, decision making, and execution times; and the netphysics of braking performances of the vehicles themselves. Ithas been shown that embedded in these models are theoreticalestimates of the speed at maximum flow, (�/�) ; maximum0.5

flow, [� + 2(� �) ] ; and the speed at which small changes in0.5 -1

traffic stream speed propagate back through a traffic stream,(�/�) (Rothery 1968). 0.5

The speed-spacing models noted above are applicable to caseswhere each vehicle in the traffic stream maintains the same ornearly the same constant speed and each vehicle is attempting to

��

� � �

maintain the same spacing (i.e., it describes a steady-state traffic 1958 by Herman and his associates at the General Motorsstream). Research Laboratories. These research efforts were microscopic

Through the work of Reuschel (1950) and Pipes (1953), the which one vehicle followed another. With such a description,dynamical elements of a line of vehicles were introduced. In the macroscopic behavior of single lane traffic flow can bethese works, the focus was on the dynamical behavior of a approximated. Hence, car following models form a bridgestream of vehicles as they accelerate or decelerate and each between individual "car following" behavior and thedriver-vehicle pair attempts to follow one another. These efforts macroscopic world of a line of vehicles and their correspondingwere extended further through the efforts of Kometani and flow and stability properties.Sasaki (1958) in Japan and in a series of publications starting in

approaches that focused on describing the detailed manner in

4.1 Model Development

Car following models of single lane traffic assume that there is vehicle characteristics or class ofa correlation between vehicles in a range of inter-vehicle characteristics and from the driver's vastspacing, from zero to about 100 to 125 meters and provides an repertoire of driving experience. Theexplicit form for this coupling. The modeling assumes that each integration of current information anddriver in a following vehicle is an active and predictable control catalogued knowledge allows for theelement in the driver-vehicle-road system. These tasks are development of driving strategies whichtermed psychomotor skills or perceptual-motor skills because become "automatic" and from which evolvethey require a continued motor response to a continuous series "driving skills".of stimuli.

The relatively simple and common driving task of one vehicle commands with dexterity, smoothness, andfollowing another on a straight roadway where there is no coordination, constantly relying on feedbackpassing (neglecting all other subsidiary tasks such as steering, from his own responses which arerouting, etc.) can be categorized in three specific subtasks: superimposed on the dynamics of the system's

� Perception: The driver collects relevant informationmainly through the visual channel. This It is not clear how a driver carries out these functions in detail.information arises primarily from the motion The millions of miles that are driven each year attest to the factof the lead vehicle and the driver's vehicle. that with little or no training, drivers successfully solve aSome of the more obvious information multitude of complex driving tasks. Many of the fundamentalelements, only part of which a driver is questions related to driving tasks lie in the area of 'humansensitive to, are vehicle speeds, accelerations factors' and in the study of how human skill is related toand higher derivatives (e.g., "jerk"), inter- information processes. vehicle spacing, relative speeds, rate ofclosure, and functions of these variables (e.g., The process of comparing the inputs of a human operator to thata "collision time"). operator's outputs using operational analysis was pioneered by

� Decision These attempts to determine mathematical expressions linkingMaking: A driver interprets the information obtained by input and output have met with limited success. One of the

sampling and integrates it over time in order to primary difficulties is that the operator (in our case the driver)provide adequate updating of inputs. has no unique transfer function; the driver is a differentInterpreting the information is carried out 'mechanism' under different conditions. While such an approachwithin the framework of a knowledge of has met with limited success, through the course of studies like

� Control: The skilled driver can execute control

counterparts (lead vehicle and roadway).

the work of Tustin (1947), Ellson (1949), and Taylor (1949).

<U��Uf> � <Urel>

1�t�

t��t/2

t��t/2Urel(t)dt

tc �S(t)Urel

��

� � �

(4.5)

(4.6)

these a number of useful concepts have been developed. For where � is a proportionality factor which equates the stimulusexample, reaction times were looked upon as characteristics of function to the response or control function. The stimulusindividuals rather than functional characteristics of the task itself. function is composed of many factors: speed, relative speed, In addition, by introducing the concept of "information", it has inter-vehicle spacing, accelerations, vehicle performance, driverproved possible to parallel reaction time with the rate of coping thresholds, etc. with information.

The early work by Tustin (1947) indicated maximum rates of the question is, which of these factors are the most significant fromorder of 22-24 bits/second (sec). Knowledge of human an explanatory viewpoint. Can any of them be neglected and stillperformance and the rates of handling information made it retain an approximate description of the situation beingpossible to design the response characteristics of the machine for modeled? maximum compatibility of what really is an operator-machinesystem. What is generally assumed in car following modeling is that a

The very concept of treating an operator as a transfer function avoid collisions.implies, partly, that the operator acts in some continuousmanner. There is some evidence that this is not completely These two elements can be accomplished if the driver maintainscorrect and that an operator acts in a discontinuous way. Thereis a period of time during which the operator having made a"decision" to react is in an irreversible state and that the responsemust follow at an appropriate time, which later is consistent withthe task.

The concept of a human behavior being discontinuous incarrying out tasks was first put forward by Uttley (1944) andhas been strengthened by such studies as Telfor's (1931), whodemonstrated that sequential responses are correlated in such away that the response-time to a second stimulus is affectedsignificantly by the separation of the two stimuli. Inertia, on theother hand, both in the operator and the machine, creates anappearance of smoothness and continuity to the control element.

In car following, inertia also provides direct feedback data to theoperator which is proportional to the acceleration of the vehicle.Inertia also has a smoothing effect on the performancerequirements of the operator since the large masses and limitedoutput of drive-trains eliminate high frequency components ofthe task.

Car following models have not explicitly attempted to take all ofthese factors into account. The approach that is used assumesthat a stimulus-response relationship exists that describes, atleast phenomenologically, the control process of a driver-vehicleunit. The stimulus-response equation expresses the concept thata driver of a vehicle responds to a given stimulus according to arelation:

Response = � Stimulus (4.4)

Do all of these factors come into play part of the time? The

driver attempts to: (a) keep up with the vehicle ahead and (b)

a small average relative speed, U over short time periods, sayrel

�t, i.e.,

is kept small. This ensures that ‘collision’ times:

are kept large, and inter-vehicle spacings would not appreciablyincrease during the time period, �t. The duration of the �t willdepend in part on alertness, ability to estimate quantities such as:spacing, relative speed, and the level of information required forthe driver to assess the situation to a tolerable probability level(e.g., the probability of detecting the relative movement of anobject, in this case a lead vehicle) and can be expressed as afunction of the perception time.

Because of the role relative-speed plays in maintaining relativelylarge collision times and in preventing a lead vehicle from'drifting' away, it is assumed as a first approximation that theargument of the stimulus function is the relative speed.

From the discussion above of driver characteristics, relativespeed should be integrated over time to reflect the recent timehistory of events, i.e., the stimulus function should have the formlike that of Equation 4.5 and be generalized so that the stimulus

< U��Uf > � <Urel >��

t��t/2

t��t/2�(t�t �)Urel(t

�)dt

�(t) � �(t�T)

Past

Weightingfunction

Future

Now

Time

�(t�T) � 0, for t�T

�(t�T) � 1, for t � T

��

o�(t�T)dt � 1

T(t) ��t

0t ��(t �)dt �

T

��

� � �

(4.7)

(4.8)

(4.9)

(4.10)

(4.12)

at a given time, t, depends on the weighted sum of all earliervalues of the relative speed, i.e.,

where � (t) is a weighing function which reflects a driver'sestimation, evaluation, and processing of earlier information(Chandler et al. 1958). The driver weighs past and presentinformation and responds at some future time. The consequenceof using a number of specific weighing functions has beenexamined (Lee 1966), and a spectral analysis approach has beenused to derive a weighing function directly from car followingdata (Darroch and Rothery 1969).

The general features of a weighting function are depicted inFigure 4.1. What has happened a number of seconds (� 5 sec)in the past is not highly relevant to a driver now, and for a shorttime (� 0.5 sec) a driver cannot readily evaluate the information which corresponds to a simple constant response time, T, for aavailable to him. One approach is to assume that

where

and

For this case, our stimulus function becomes

Stimulus(t) = U (t - T) - U (t - T) (4.11)� f

driver-vehicle unit. In the general case of � (t), there is anaverage response time, , given by

Figure 4.1Schematic Diagram of Relative Speed Stimulus

and a Weighting Function Versus Time (Darroch and Rothery 1972).

xf(t) � �[x.�(t�T)�x. f(t�T)]

xf(t�T) � �[x.�(t)�x. f(t)]

x

��

� � �

(4.14)

(4.15)

The main effect of such a response time or delay is that the driveris responding at all times to a stimulus. The driver is observingthe stimulus and determining a response that will be made sometime in the future. By delaying the response, the driver obtains"advanced" information.

For redundant stimuli there is little need to delay response, apart equation of car-following, and as such it is a grossly simplifiedfrom the physical execution of the response. Redundancy alone description of a complex phenomenon. A generalization of carcan provide advance information and for such cases, response following in a conventional control theory block diagram istimes are shorter. shown in Figure 4.1a. In this same format the linear car-

The response function is taken as the acceleration of the 4.1b. In this figure the driver is represented by a time delay andfollowing vehicle, because the driver has direct control of this a gain factor. Undoubtedly, a more complete representation ofquantity through the 'accelerator' and brake pedals and also car following includes a set of equations that would model thebecause a driver obtains direct feedback of this variable through dynamical properties of the vehicle and the roadwayinertial forces, i.e., characteristics. It would also include the psychological and

Response (t) = a (t) = (t) (4.13) f f

where x (t) denotes the longitudinal position along the roadwayi

of the ith vehicle at time t. Combining Equations4.11 and 4.13into Equation 4.4 the stimulus-response equation becomes(Chandler et al. 1958):

or equivalently

Equation 4.15 is a first approximation to the stimulus-response

following model presented in Equation 4.15 is shown in Figure

physiological properties of drivers, as well as couplings betweenvehicles, other than the forward nearest neighbors and otherdriving tasks such as lateral control, the state of traffic, andemergency conditions.

For example, vehicle performance undoubtedly alters driverbehavior and plays an important role in real traffic where mixedtraffic represents a wide performance distribution, and whereappropriate responses cannot always be physically achieved bya subset of vehicles comprising the traffic stream. This is onearea where research would contribute substantially to a betterunderstanding of the growth, decay, and frequency ofdisturbances in traffic streams (see, e.g., Harris 1964; Hermanand Rothery 1967; Lam and Rothery 1970).

Figure 4.1aBlock Diagram of Car-Following.

xf(��1) � C[(x.�(�)�x. f(�))]

�xn(t) � u��v�n

��0(�1)�

t

(n��)T�

n�� (n��)T��n��1

(n�1)!�!� (u0(t��)�u)dt

��

� � �

(4.16)

Figure 4.1bBlock Diagram of the Linear Car-Following Model.

4.2 Stability Analysis

In this section we address the stability of the linear car followingequation, Equation 4.15, with respect to disturbances. Twoparticular types of stabilities are examined: local stability andasymptotic stability.

Local Stability is concerned with the response of a followingvehicle to a fluctuation in the motion of the vehicle directly infront of it; i.e., it is concerned with the localized behaviorbetween pairs of vehicles.

Asymptotic Stability is concerned with the manner in which afluctuation in the motion of any vehicle, say the lead vehicle ofa platoon, is propagated through a line of vehicles.

The analysis develops criteria which characterize the types ofpossible motion allowed by the model. For a given range ofmodel parameters, the analysis determines if the traffic stream(as described by the model) is stable or not, (i.e., whetherdisturbances are damped, bounded, or unbounded). This is animportant determination with respect to understanding theapplicability of the modeling. It identifies several characteristicswith respect to single lane traffic flow, safety, and model validity.If the model is realistic, this range should be consistent withmeasured values of these parameters in any applicable situationwhere disturbances are known to be stable. It should also beconsistent with the fact that following a vehicle is an extremelycommon experience, and is generally stable.

4.2.1 Local Stability

In this analysis, the linear car following equation, (Equation4.15) is assumed. As before, the position of the lead vehicle andthe following vehicle at a time, t, are denoted by x (t) and x (t),

� f

respectively. Rescaling time in units of the response time, T,using the transformation, t = �T, Equation 4.15 simplifies to

where C = �T. The conditions for the local behavior of thefollowing vehicle can be derived by solving Equation 4.16 by themethod of Laplace transforms (Herman et al. 1959).

The evaluation of the inverse Laplace transform for Equation4.16 has been performed (Chow 1958; Kometani and Sasaki1958). For example, for the case where the lead and followingvehicles are initially moving with a constant speed, u, thesolution for the speed of the following vehicle was given byChow where � denotes the integral part of t/T. The complexform of Chow's solution makes it difficult to describe variousphysical properties (Chow 1958).

L �1[C(C � se s)�1s

C � se s� 0

�S �

1�

(V�U)

��

0xf(t�T)dt � V�U

��

0[x. l(t)�x. f(t)]dt � �S

�S � ��

0[ �x

�(t)�x. f(t)]dt � V�U

�

e a0t e tb0t

C � e �1(�0.368), then a0�0, b0�0

��

� � �

(4.17)

(4.18)

(4.19)

(4.20)

However, the general behavior of the following vehicle's motion particular, it demonstrates that in order for the following vehiclecan be characterized by considering a specific set of initialconditions. Without any loss in generality, initial conditions areassumed so that both vehicles are moving with a constant speed,u. Then using a moving coordinate system z(t) for both the leadand following vehicles the formal solution for the accelerationof the following vehicle is given more simply by:

(4.16a)

where L denotes the inverse Laplace transform. The character-1

of the above inverse Laplace transform is determined by thesingularities of the factor (C + se ) since Cs Z (s) is a regulars -1 2

�

function. These singularities in the finite plane are the simplepoles of the roots of the equation

Similarly, solutions for vehicle speed and inter-vehicle spacingscan be obtained. Again, the behavior of the inter-vehicle spacingis dictated by the roots of Equation 4.17. Even for small t, thecharacter of the solution depends on the pole with the largest realpart, say , s = a + ib , since all other poles have considerably0 0 0larger negative real parts so that their contributions are heavilydamped.

Hence, the character of the inverse Laplace transform has theform . For different values of C, the pole with thelargest real part generates four distinct cases:

a) if , and themotion is non-oscillatory and exponentiallydamped.

b) if e < C < � / 2, then a < 0, b > 0 and the- 10 0

motion is oscillatory with exponential damping.

c) if C = � / 2 , then a = 0, b , > 0 and the motion is0 0oscillatory with constant amplitude.

d) if C > � / 2 then a > 0, b > 0 and the motion is0 0oscillatory with increasing amplitude.

The above establishes criteria for the numerical values of Cwhich characterize the motion of the following vehicle. In

not to over-compensate to a fluctuation, it is necessary that C�1/e. For values of C that are somewhat greater, oscillationsoccur but are heavily damped and thus insignificant. Dampingoccurs to some extent as long as C < �/2.

These results concerning the oscillatory and non-oscillatorybehavior apply to the speed and acceleration of the following vehicle as well as to the inter-vehicle spacing. Thus, e.g., if C �

e , the inter-vehicle spacing changes in a non-oscillatory manner-1

by the amount �S , where

when the speeds of the vehicle pair changes from U to V. Animportant case is when the lead vehicle stops. Then, the finalspeed, V, is zero, and the total change in inter-vehicle spacing is- U/ �.

In order for a following vehicle to avoid a 'collision' frominitiation of a fluctuation in a lead vehicle's speed the inter-vehicle spacing should be at least as large as U/�. On the otherhand, in the interests of traffic flow the inter-vehicle spacingshould be small by having � as large as possible and yet withinthe stable limit. Ideally, the best choice of � is (eT) .-1

The result expressed in Equation 4.18 follows directly fromChow's solution (or more simply by elementary considerations).Because the initial and final speeds for both vehicles are U andV, respectively, we have

and from Equation 4.15 we have

or

��

� � �

as given earlier in Equation 4.18.

In order to illustrate the general theory of local stability, theresults of several calculations using a Berkeley Ease analogcomputer and an IBM digital computer are described. It isinteresting to note that in solving the linear car followingequation for two vehicles, estimates for the local stabilitycondition were first obtained using an analog computer fordifferent values of C which differentiate the various type ofmotion.

Figure 4.2 illustrates the solutions for C= e , where the lead-1

vehicle reduces its speed and then accelerates back to its originalspeed. Since C has a value for the locally stable limit, theacceleration and speed of the following vehicle, as well as theinter-vehicle spacing between the two vehicles are non-oscillatory.

In Figure 4.3, the inter-vehicle spacing is shown for four othervalues of C for the same fluctuation of the lead vehicle as shownin Figure 4.2. The values of C range over the cases of oscillatory

Note: Vehicle 2 follows Vehicle 1 (lead car) with a time lag T=1.5 sec and a value of C=e (�0.368), the limiting value for local-1

stability. The initial velocity of each vehicle is u

Figure 4.2Detailed Motion of Two Cars Showing the

Effect of a Fluctuation in the Acceleration of the Lead Car (Herman et al. 1958).

xf(��1) � C d m

dt m[x

�(�)�xf(�)]

C� s me s� 0

�

�

2

��

� � �

(4.21)

(4.22)

Note: Changes in car spacings from an original constant spacing between two cars for the noted values of C. The a ccelerationprofile of the lead car is the same as that shown in Figure 4.2.

Figure 4.3Changes in Car Spacings from an

Original Constant Spacing Between Two Cars (Herman et al. 1958).

motion where the amplitude is damped, undamped, and ofincreasing amplitude.

For the values of C = 0.5 and 0.80, the spacing is oscillatory andheavily damped.

For C = 1.57 ( ), Equation 4.21 is

the spacing oscillates with constant amplitude. For C = 1.60, themotion is oscillatory with increasing amplitude.

Local Stability with Other Controls. Qualitative arguments canbe given of a driver's lack of sensitivity to variation in relativeacceleration or higher derivatives of inter-vehicle spacingsbecause of the inability to make estimates of such quantities. Itis of interest to determine whether a control centered aroundsuch derivatives would be locally stable. Consider the carfollowing equation of the form

for m= 0,1,2,3..., i.e., a control where the acceleration of thefollowing vehicle is proportional to the mth derivative of the

inter-vehicle spacing. For m = 1, we obtain the linear carfollowing equation.

Using the identical analysis for any m, the equation whose rootsdetermine the character of the motion which results from

None of these roots lie on the negative real axis when m is even,therefore, local stability is possible only for odd values of themth derivative of spacing: relative speed, the first derivative ofrelative acceleration (m = 3), etc. Note that this result indicatesthat an acceleration response directly proportional to inter-vehicle spacing stimulus is unstable.

4.2.2 Asymptotic Stability

In the previous analysis, the behavior of one vehicle followinganother was considered. Here a platoon of vehicles (except forthe platoon leader) follows the vehicle ahead according to the

xn�1(t�T) � �[ �xn(t)� �xn�1(t)]

uo(t) � ao� foe i�t

un(t) � ao� fn e i�t

un(t) � ao�F(�,�,�,n)e i�(�,�,�,n)

[1�(��

)2�2(�

�)sin(��)]�n/2

1�(��

)2�2(�

�)sin(��) > 1

�

�> 2sin(��)

�� < 12

[lim��0(��)/sin(��)]

xn�1(t)�u0(t)4�n�

12�

�T �½

� exp [t�n/�]4n/�(1/2��)

��

� � ��

(4.23)

(4.24)

(4.25)

(4.26)

(4.27)

(4.28)

linear car following equation, Equation 4.15. The criterianecessary for asymptotic stability or instability were firstinvestigated by considering the Fourier components of the speedfluctuation of a platoon leader (Chandler et al. 1958).

The set of equations which attempts to describe a line of N i.e. ifidentical car-driver units is:

where n =0,1,2,3,...N.

Any specific solution to these equations depends on the velocityprofile of the lead vehicle of the platoon, u (t), and the two0

parameters � and T. For any inter-vehicle spacing, if adisturbance grows in amplitude then a 'collision' wouldeventually occur somewhere back in the line of vehicles.

While numerical solutions to Equation 4.23 can determine atwhat point such an event would occur, the interest is todetermine criteria for the growth or decay of such a disturbance.Since an arbitrary speed pattern can be expressed as a linearcombination of monochromatic components by Fourier analysis,the specific profile of a platoon leader can be simply representedby one component, i.e., by a constant together with amonochromatic oscillation with frequency, � and amplitude, fo, i.e.,

and the speed profile of the nth vehicle by

Substitution of Equations 4.24 and 4.25 into Equation 4.23yields:

where the amplitude factor F (�, �, �, n) is given by

which decreases with increasing n if

The severest restriction on the parameter � arises from the lowfrequency range, since in the limit as � � 0, � must satisfy theinequality

Accordingly, asymptotic stability is insured for all frequencieswhere this inequality is satisfied.

For those values of � within the physically realizable frequencyrange of vehicular speed oscillations, the right hand side of theinequality of 4.27 has a short range of values of 0.50 to about0.52. The asymptotic stability criteria divides the two parameterdomain into stable and unstable regions, as graphicallyillustrated in Figure 4.4.

The criteria for local stability (namely that no local oscillationsoccur when �� e ) also insures asymptoticstability. It has also-1

been shown (Chandler et al. 1958) that a speed fluctuation canbe approximated by:

Hence, the speed of propagation of the disturbance with respectto the moving traffic stream in number of inter-vehicleseparations per second, n/t, is �.

That is, the time required for the disturbance to propagatebetween pairs of vehicles is � , a constant, which is independent-1

of the response time T. It is noted from the above equation thatin the propagation of a speed fluctuation the amplitude of the

c=0.52

Unstable

Stable

1.00

0.75

0.50

0.25

0 0.5 1.0 1.5 2.0

T (sec)

c=0.50

��

� � ��

Figure 4.4Regions of Asymptotic Stability (Rothery 1968).

disturbance grows as the response time, T, approaches 1/(2�) until instability is reached. Thus, while�T < 0.5 ensures stability, short reaction times increase the rangeof the sensitivity coefficient, �, that ensures stability. From apractical viewpoint, small reaction times also reduce relativelylarge responses to a given stimulus, or in contrast, largerresponse times require relatively large responses to a givenstimulus. Acceleration fluctuations can be correspondinglyanalyzed (Chandler et al. 1958).

4.2.1.1 Numerical Examples

In order to illustrate the general theory of asymptotic stability asoutlined above, the results of a number of numerical calculationsare given. Figure 4.5 graphically exhibits the inter-vehiclespacings of successive pairs of vehicles versus time for a platoonof vehicles. Here, three values of C were used: C = 0.368, 0.5,and 0.75. The initial fluctuation of the lead vehicle, n = 1, wasthe same as that of the lead vehicle illustrated in Figure 4.2. Thisdisturbance consists of a slowing down and then a speeding upto the original speed so that the integral of acceleration over timeis zero. The particularly stable, non-oscillatory response isevident in the first case where C = 0.368 (�1/e), the localstability limit. As analyzed, a heavily damped oscillation occursin the second case where C = 0.5, the asymptotic limit. Note thatthe amplitude of the disturbance is damped as it propagates

through the line of vehicles even though this case is at theasymptotic limit.

This results from the fact that the disturbance is not a singleFourier component with near zero frequency. However,instability is clearly exhibited in the third case of Figure 4.5where C = 0.75 and in Figure 4.6 where C = 0.8. In the caseshown in Figure 4.6, the trajectories of each vehicle in a platoonof nine are graphed with respect to a coordinate system movingwith the initial platoon speed u. Asymptotic instability of aplatoon of nine cars is illustrated for the linear car followingequation, Equation 4.23, where C = 0.80. For t = 0, the vehiclesare all moving with a velocity u and are separated by a distanceof 12 meters. The propagation of the disturbance, which can bereadily discerned, leads to "collision" between the 7th and 8thcars at about t = 24 sec. The lead vehicle at t = 0 deceleratesfor 2 seconds at 4 km/h/sec, so that its speed changes from u tou -8 km/h and then accelerates back to u. This fluctuation in thespeed of the lead vehicle propagates through the platoon in anunstable manner with the inter-vehicle spacing between theseventh and eighth vehicles being reduced to zero at about 24.0sec after the initial phase of the disturbance is generated by thelead vehicle of the platoon.

In Figure 4.7 the envelope of the minimum spacing that occursbetween successive pairs of vehicles is graphed versus time

xn�2(t��) � �1[ �xn�1(t)� �xn�2(t)]��2[ �xn(t)� �xn�2]

��

� � ��

(4.29)

Note: Diagram uses Equation 4.23 for three values of C. The fluctuation in acceleration of the lead car, car number 1, is thesame as that shown in Fig. 4.2 At t=0 the cars are separated by a spacing of 21 meters.

Figure 4.5Inter-Vehicle Spacings of a Platoon of Vehicles

Versus Time for the Linear Car Following Model (Herman et al. 1958).

where the lead vehicle's speed varies sinusoidally with a point. Here, the numerical solution yields a maximum andfrequency � =2�/10 radian/sec. The envelope of minimum minimum amplitude that is constant to seven significant places.inter-vehicle spacing versus vehicle position is shown for threevalues of �. The response time, T, equals 1 second. It has beenshown that the frequency spectrum of relative speed andacceleration in car following experiments have essentially alltheir content below this frequency (Darroch and Rothery 1973).

The values for the parameter � were 0.530, 0.5345, and0.550/sec. The value for the time lag, T, was 1 sec in each case.The frequency used is that value of � which just satisfiesthe stability equation, Equation 4.27, for the case where�= 0.5345/sec. This latter figure serves to demonstrate not onlythe stability criteria as a function of frequency but the accuracyof the numerical results. A comparison between that which ispredicted from the stability analysis and the numerical solutionfor the constant amplitude case (�=0.5345/sec) serves as a check

4.2.1.2 Next-Nearest Vehicle Coupling

In the nearest neighbor vehicle following model, the motion ofeach vehicle in a platoon is determined solely by the motion ofthe vehicle directly in front. The effect of including the motionof the "next nearest neighbor" vehicle (i.e., the car which is twovehicles ahead in addition to the vehicle directly in front) can beascertained. An approximation to this type of control, is themodel

��

� � ��

Note: Diagram illustrates the linear car following equation, eq. 4.23, where C=080.

Figure 4.6Asymptotic Instability of a Platoon of Nine Cars (Herman et al. 1958).

Figure 4.7Envelope of Minimum Inter-Vehicle Spacing Versus Vehicle Position (Rothery 1968).

(�1��2)� < 12

(��)/sin(��)] (�1��2)� > 12

xn�1(t��) � �[ �xn(t)� �xn�1(t)]

Sf � Si��1(Uf � Ui)

k �1f � k �1

i ��1(Uf�Ui)

��

� � ��

(4.30) (4.31)

(4.32)

(4.33)

(4.34)

which in the limit �� 0 is This equation states that the effect of adding next nearestneighbor coupling to the control element is, to the first order, toincrease � to (� + � ). This reduces the value that � can have1 1 2 1and still maintain asymptotic stability.

4.3 Steady-State Flow

This section discusses the properties of steady-state traffic flow also follows from elementary considerations by integration ofbased on car following models of single-lane traffic flow. In Equation 4.32 as shown in the previous section (Gazis et al.particular, the associated speed-spacing or equivalent speed-concentration relationships, as well as the flow-concentrationrelationships for single lane traffic flow are developed.

The Linear Case. The equations of motion for a single lane oftraffic described by the linear car following model are given by:

where n = 1, 2, 3, ....

In order to interrelate one steady-state to another under thiscontrol, assume (up to a time t=0) each vehicle is traveling at a 1) They link an initial steady-state to a second arbitraryspeed U and that the inter-vehicle spacing is S . Suppose thati i

at t=0, the lead vehicle undergoes a speed change and increasesor decreases its speed so that its final speed after some time, t, isU . A specific numerical solution of this type of transition isfexhibited in Figure 4.8.

In this example C = �T=0.47 so that the stream of traffic isstable, and speed fluctuations are damped. Any case where theasymptotic stability criteria is satisfied assures that eachfollowing vehicle comprising the traffic stream eventuallyreaches a state traveling at the speed U . In the transition fromf

a speed U to a speed U , the inter-vehicle spacing S changesi f

from S to S , wherei f

This result follows directly from the solution to the car followingequation, Equation 4.16a or from Chow (1958). Equation 4.33

1959). This result is not directly dependent on the time lag, T,except that for this result to be valid the time lag, T, must allowthe equation of motion to form a stable stream of traffic. Sincevehicle spacing is the inverse of traffic stream concentration, k,the speed-concentration relation corresponding to Equation 4.33is:

The significance of Equations 4.33 and 4.34 is that:

steady-state, and

2) They establish relationships between macroscopic trafficstream variables involving a microscopic car followingparameter, � .

In this respect they can be used to test the applicability of the carfollowing model in describing the overall properties of singlelane traffic flow. For stopped traffic, U = 0, and thei

corresponding spacing, S , is composed of vehicle length ando"bumper-to-bumper" inter-vehicle spacing. The concentrationcorresponding to a spacing, S , is denoted by k and is frequentlyo jreferred to as the 'jam concentration'.

Given k, then Equation 4.34 for an arbitrary traffic state definedj

by a speed, U, and a concentration, k, becomes

U � �(k �1�k �1

j )

q � Uk � �(1� kkj

)

��

� � ��

(4.35)

(4.36)

Note: A numerical solution to Equation 4.32 for the inter-vehicle spacings of an 11- vehicle platoon going from one steady-sta te toanother (�T = 0.47). The lead vehicle's speed decreases by 7.5 meters per second.

Figure 4.8Inter-Vehicle Spacings of an Eleven Vehicle Platoon (Rothery 1968).

A comparison of this relationship was made (Gazis et al. 1959)with a specific set of reported observations (Greenberg 1959) fora case of single lane traffic flow (i.e., for the northbound trafficflowing through the Lincoln Tunnel which passes under the The inability of Equation 4.36 to exhibit the required qualitativeHudson River between the States of New York and New Jersey). relationship between flow and concentration (see Chapter 2) ledThis comparison is reproduced in Figure 4.9 and leads to an to the modification of the linear car following equation (Gazis etestimate of 0.60 sec for �. This estimate of � implies an upper al. 1959).-1

bound for T � 0.83 sec for an asymptotic stable traffic streamusing this facility.

While this fit and these values are not unreasonable, afundamental problem is identified with this form of an equationfor a speed-spacing relationship (Gazis et al. 1959). Because itis linear, this relationship does not lead to a reasonabledescription of traffic flow. This is illustrated in Figure 4.10where the same data from the Lincoln Tunnel (in Figure 4.9) isregraphed. Here the graph is in the form of a normalized flow,

versus a normalized concentration together with thecorresponding theoretical steady-state result derived fromEquation 4.35, i.e.,

Non-Linear Models. The linear car following model specifiesan acceleration response which is completely independent ofvehicle spacing (i.e., for a given relative velocity, response is thesame whether the vehicle following distance is small [e.g., of theorder of 5 or 10 meters] or if the spacing is relatively large [i.e.,of the order of hundreds of meters]). Qualitatively, we wouldexpect that response to a given relative speed to increase withsmaller spacings.

��

� � ��

Note: The data are those of (Greenberg 1959) for the Lincoln Tunnel. The curve represents a "least squares fit" of Equation 4.35to the data.

Figure 4.9Speed (miles/hour) Versus Vehicle Concentration (vehicles/mile).(Gazis et al. 1959).

Note: The curve corresponds to Equation 4.36 where the parameters are those from the "fit" shown in Figure 4.9.

Figure 4.10Normalized Flow Versus Normalized Concentration (Gazis et al. 1959).

� � �1 /S(t) � �1 /[xn(t)�xn�1(t)]

xn�1(t��) ��1

[xn(t)�xn�1(t)][ �xn(t)� �xn�1(t)]

u � �1ln (kj /k)

q � �1kln(kj /k)

��

� � ��

(4.37)

(4.38)

(4.39)

(4.40)

In order to attempt to take this effect into account, the linear therefore the flow-concentration relationship, does not describemodel is modified by supposing that the gain factor, �, is not a the state of the traffic stream. constant but is inversely proportional to vehicle spacing, i.e.,

where � is a new parameter - assumed to be a constant and approaches to single-lane traffic flow because in these cases any1which shall be referred to as the sensitivity coefficient. Using small speed change, once the disturbance arrives, each vehicleEquation 4.37 in Equation 4.32, our car following equation is: instantaneously relaxes to the new speed, at the 'proper' spacing.

for n = 1,2,3,... downstream from the vehicle initiating the speed change at a

As before, by assuming the parameters are such that the trafficstream is stable, this equation can be integrated yielding thesteady-state relation for speed and concentration:

and for steady-state flow and concentration:

where again it is assumed that for u=0, the spacing is equal toan effective vehicle length, L = k . These relations for steady--1

state flow are identical to those obtained from considering thetraffic stream to be approximated by a continuous compressiblefluid (see Chapter 5) with the property that disturbances arepropagated with a constant speed with respect to the movingmedium (Greenberg 1959). For our non-linear car followingequation, infinitesimal disturbances are propagated with speed� . This is consistent with the earlier discussion regarding the1speed of propagation of a disturbance per vehicle pair.

It can be shown that if the propagation time, � , is directly0

proportional to spacing (i.e., � � S), Equations 4.39 and 4.400

are obtained where the constant ratio S /� is identified as the Assuming that this data is a representative sample of thisoconstant � . facility's traffic, the value of 27.7 km/h is an estimate not only ofl

These two approaches are not analogous. In the fluid analogy but it is the 'characteristic speed' for the roadway undercase, the speed-spacing relationship is 'followed' at every instant consideration (i.e., the speed of the traffic stream whichbefore, during, and after a disturbance. In the case of car maximizes the flow). following during the transition phase, the speed-spacing, and

A solution to any particular set of equations for the motion of atraffic stream specifies departures from the steady-state. This isnot the case for simple headway models or hydro-dynamical

This emphasizes the shortcoming of these alternate approaches.They cannot take into account the behavioral and physicalaspects of disturbances. In the case of car following models, theinitial phase of a disturbance arrives at the nth vehicle

time (n-1)T seconds after the onset of the fluctuation. The timeit takes vehicles to reach the changed speed depends on theparameter �, for the linear model, and � , for the non-linear1

model, subject to the restriction that � > T or � < S/T,-11

respectively.

These restrictions assure that the signal speed can never precedethe initial phase speed of a disturbance. For the linear case, therestriction is more than satisfied for an asymptotic stable traffic stream. For small speed changes, it is also satisfied for the non-linear model by assuming that the stability criteria results for thelinear case yields a bound for the stability in the non-linear case. Hence, the inequality �� /S*<0.5 provides a sufficient stabilitycondition for the non-linear case, where S* is the minimumspacing occurring during a transition from one steady-state toanother.

Before discussing a more general form for the sensitivitycoefficient (i.e., Equation 4.37), the same reported data(Greenberg 1959) plotted in Figures 4.9 and 4.10 are graphed inFigures 4.11 and 4.12 together with the steady-state relations(Equations 4.39 and 4.40 obtained from the non-linear model,Equation 4.38). The fit of the data to the steady-state relation viathe method of "least squares" is good and the resulting values for� and k are 27.7 km/h and 142 veh/km, respectively.1 j

the sensitivity coefficient for the non-linear car following model

��

� � ��

Note: The curve corresponds to a "least squares" fit of Equation 4.39 to the data (Greenberg 1959).

Figure 4.11Speed Versus Vehicle Concentration (Gazis et al. 1959).

Note: The curve corresponds to Equation 4.40 where parameters are those from the "fit" obtained in Figure 4.11.

Figure 4.12Normalized Flow Versus Normalized Vehicle Concentration (Edie et al. 1963).

� � �2x.n�1(t��)/[xn(t)�xn�1(t)]

2

xn�1(t��)��2x

.n�1(t��)

[xn(t)�xn�1(t)]2[x. n(t)�x.n�1(t)]

U � Uf e�k/km

q � Uf ke �k/km

U � Uf for 0�k�kf

U � Uf exp �k � kf

km

U � Uf (1�k/kj)

U � Uf (1�L/S)

�U � (Uf L/S 2) �S

xn�1(t��) �Uf L

[xn(t)�xn�1(t)]2[ �xn(t)� �xn�1(t)]

��

� � ��

(4.41)

(4.42)

(4.43)

(4.44)

(4.45)

(4.46)

(4.47)

(4.48)

(4.49)

The corresponding vehicle concentration at maximum flow, i.e.,when u = � , is e k . This predicts a roadway capacity of1 j

-l

� e k of about �1400 veh/h for the Lincoln Tunnel. A noted1 j-l

undesirable property of Equation 4.40 is that the tangent dq/dtis infinite at k = 0, whereas a linear relation between flow andconcentration would more accurately describe traffic near zeroconcentration. This is not a serious defect in the model since carfollowing models are not applicable for low concentrationswhere spacings are large and the coupling between vehicles areweak. However, this property of the model did suggest thefollowing alternative form (Edie 1961) for the gain factor,

This leads to the following expression for a car following model:

As before, this can be integrated giving the following steady-state equations:

and

where U is the "free mean speed", i.e., the speed of the trafficf

stream near zero concentration and k is the concentration whenmthe flow is a maximum. In this case the sensitivity coefficient, �2

can be identified as k . The speed at optimal flow is e Um f-1 -1

which, as before, corresponds to the speed of propagation of adisturbance with respect to the moving traffic stream. Thismodel predicts a finite speed, U , near zero concentration. f

Ideally, this speed concentration relation should be translated tothe right in order to more completely take into accountobservations that the speed of the traffic stream is independentof vehicle concentration for low concentrations, .i.e.

and

where k corresponds to a concentration where vehicle tofvehicle interactions begin to take place so that the stream speedbegins to decrease with increasing concentration. Assuming thatinteractions take place at a spacing of about 120 m, k wouldfhave a value of about 8 veh/km. A "kink" of this kind wasintroduced into a linear model for the speed concentrationrelationship (Greenshields 1935).

Greenshields' empirical model for a speed-concentration relationis given by

where U is a “free mean speed” and k is the jam concentration.f j

It is of interest to question what car following model wouldcorrespond to the above steady-state equations as expressed byEquation 4.46. The particular model can be derived in thefollowing elementary way (Gazis et al. 1961). Equation 4.46 isrewritten as

Differentiating both sides with respect to time obtains

which after introduction of a time lag is for the (n+1) vehicle:

Uf L

[xn(t)�xn�1(t)]2

xn�1(t��) � �[ �xn(t)� �xn�1(t)]

� � a�,m �x m

n�1(t��)/[xn(t)�xn�1(t)]�

fm(U) � a � fl(S)�b

fp(x) � x 1�p

fp(x) � �nx

b � fm(Uf )

b � �afl(L)

��

� � ��

(4.50)

(4.51)

(4.52)

(4.53)

(4.54)

(4.55)

(4.56)

(4.57)

The gain factor is:

The above procedure demonstrates an alternate technique atarriving at stimulus response equations from relativelyelementary considerations. This method was used to developearly car following models (Reuschel 1950; Pipes 1951). Thetechnique does pre-suppose that a speed-spacing relation reflectsdetailed psycho-physical aspects of how one vehicle followsanother. To summarize the car-following equation considered,we have:

where the factor, �, is assumed to be given by the following:

� A constant, � = � ;0

� A term inversely proportional to the spacing, � = � /S;1� A term proportional to the speed and inversely

proportional to the spacing squared, � = � U/S ; and22

� A term inversely proportional to the spacing squared, � = � / S .3

2

These models can be considered to be special cases of a moregeneral expression for the gain factor, namely:

where a is a constant to be determined experimentally. Model�,m

specification is to be determined on the basis of the degree towhich it presents a consistent description of actual trafficphenomena. Equations 4.51 and 4.52 provide a relatively broadframework in so far as steady-state phenomena is concerned(Gazis et al. 1961).

Using these equations and integrating over time we have

where, as before, U is the steady-state speed of the traffic stream,S is the steady-state spacing, and a and b are appropriate

constants consistent with the physical restrictions and wheref (x), (p = m or �), is given by p

for p � 1 and

for p = 1. The integration constant b is related to the "free meanspeed" or the "jam concentration" depending on the specificvalues of m and �. For m > 1, �� 1, or m =1, � >1

and

for all other combinations of m and �, except � < 1 and m = 1.

For those cases where � < 1 and m = 1 it is not possible to satisfyeither the boundary condition at k = 0 or k and the integrationj

constant can be assigned arbitrarily, e.g., at k , the concentrationmat maximum flow or more appropriately at some 'critical'concentration near the boundary condition of a "free speed"determined by the "kink" in speed-concentration data for theparticular facility being modeled. The relationship between km

and k is a characteristic of the particular functional or modeljbeing used to describe traffic flow of the facility being studiedand not the physical phenomenon involved. For example, for thetwo models given by � = 1, m = 0, and � = 2, m = 0, maximumflow occurs at a concentration of e k and k / 2 , respectively.-l

j jSuch a result is not physically unrealistic. Physically thequestion is whether or not the measured value of q occurs atmax

or near the numerical value of these terms, i.e., k = e k or k /2m j j-1

for the two examples cited.

Using Equations 4.53, 4.54, 4.55, 4.56, 4.57, and the definitionof steady-state flow, we can obtain the relationships betweenspeed, concentration, and flow. Several examples have beengiven above. Figures 4.13 and 4.14 contain these and additionalexamples of flow versus concentration relations for various

��

� � ��

Note: Normalized flow versus normalized concentration corresponding to the steady-state solution of Equations 4.51 and 4.52for m=1 and various values of �.

Figure 4.13Normalized Flow Versus Normalized Concentration (Gazis et al. 1963).

Figure 4.14Normalized Flow versus Normalized Concentration Corresponding to the Steady-State

Solution of Equations 4.51 and 4.52 for m=1 and Various Values of �� (Gazis 1963).

��

� � ��

values of � and m. These flow curves are normalized by lettingq = q/q , and k = k/k .n max n j

It can be seen from these figures that most of the models shownhere reflect the general type of flow diagram required to agreewith the qualitative descriptions of steady-state flow. Thespectrum of models provided are capable of fitting data like thatshown in Figure 4.9 so long as a suitable choice of theparameters is made. spacing, i.e., m= 0, � =1 (Herman et al. 1959). A generalized

equation for steady-state flow (Drew 1965) and subsequentlyThe generalized expression for car following models, Equations tested on the Gulf Freeway in Houston, Texas led to a model4.51 and 4.52, has also been examined for non-integral valuesfor m and � (May and Keller 1967). Fittingdata obtained on theEisenhower Expressway in Chicago they proposed a model with As noted earlier, consideration of a "free-speed" near lowm = 0.8 and � = 2.8. Various values for m and � can be identifiedin the early work on steady-state flow and car following . model m = 1, � = 2 (Edie 1961). Yet another model, m = 1,

The case m = 0, � = 0 equates to the "simple" linear car followingmodel. The case m = 0, � = 2 can be identified with a modeldeveloped from photographic observations of traffic flow madein 1934 (Greenshields 1935). This model can also be developed

considering the perceptual factors that are related to the carfollowing task (Pipes and Wojcik 1968; Fox and Lehman 1967;Michaels 1963). As was discussed earlier, the case for m = 0,� = 1 generates a steady-state relation that can be developed byafluid flow analogy to traffic (Greenberg 1959) and led to thereexamination of car following experiments and the hypothesisthat drivers do not have a constant gain factor to a given relative-speed stimulus but rather that it varies inversely with the vehicle

where m = 0 and � = 3/2.

concentrations led to the proposal and subsequent testing of the

� = 3 resulted from analysis of data obtained on the EisenhowerExpressway in Chicago (Drake et al. 1967). Further analysis ofthis model together with observations suggest that the sensitivitycoefficient may take on different values above a lane flow ofabout 1,800 vehicles/hr (May and Keller 1967).

4.4 Experiments And Observations

This section is devoted to the presentation and discussion of derived from car following models for steady-state flow areexperiments that have been carried out in an effort to ascertain examined. whether car following models approximate single lane trafficcharacteristics. These experiments are organized into two Finally, the degree to which any specific model of the typedistinct categories. examined in the previous section is capable of representing a

The first of these is concerned with comparisons between car macroscopic viewpoints is examined. following models and detailed measurements of the variablesinvolved in the driving situation where one vehicle followsanother on an empty roadway. These comparisons lead to aquantitative measure of car following model estimates for thespecific parameters involved for the traffic facility and vehicletype used.

The second category of experiments are those concerned withthe measurement of macroscopic flow characteristics: the studyof speed, concentration, flow and their inter-relationships forvehicle platoons and traffic environments where traffic ischanneled in a single lane. In particular, the degree to which thistype of data fits the analytical relationships that have been

consistent framework from both the microscopic and

4.4.1 Car Following Experiments

The first experiments which attempted to make a preliminaryevaluation of the linear car following model were performed anumber of decades ago (Chandler et al. 1958; Kometani andSasaki 1958). In subsequent years a number of different testswith varying objectives were performed using two vehicles,three vehicles, and buses. Most of these tests were conducted ontest track facilities and in vehicular tunnels.

xn�1(t��) � �[ �xn(t)� �xn�1(t)]��xn(t)

��

� � ��

(4.58)

In these experiments, inter-vehicle spacing, relative speed, speedof the following vehicle, and acceleration of the followingvehicles were recorded simultaneously together with a clocksignal to assure synchronization of each variable with everyother.

These car following experiments are divided into six specificcategories as follows:

1) Preliminary Test Track Experiments. The firstexperiments in car following were performed by (Chandleret al. 1958) and were carried out in order to obtainestimates of the parameters in the linear car followingmodel and to obtain a preliminary evaluation of this model.Eight male drivers participated in the study which wasconducted on a one-mile test track facility.

2) Vehicular Tunnel Experiments. To further establish the even though the relative speed has been reduced to zero orvalidity of car following models and to establish estimates, near zero. This situation was observed in several cases inthe parameters involved in real operating environments tests carried out in the vehicular tunnels - particularlywhere the traffic flow characteristics were well known, a when vehicles were coming to a stop. Equation 4.58series of experiments were carried out in the Lincoln, above allows for a non-zero acceleration when the relativeHolland, and Queens Mid-Town Tunnels of New York speed is zero. A value of � near one would indicate anCity. Ten different drivers were used in collecting 30 test attempt to nearly match the acceleration of the lead driverruns. for such cases. This does not imply that drivers are good

3) Bus Following Experiments. A series of experimentswere performed to determine whether the dynamicalcharacteristics of a traffic stream changes when it iscomposed of vehicles whose performance measures aresignificantly different than those of an automobile. Theywere also performed to determine the validity and measureparameters of car following models when applied to heavyvehicles. Using a 4 kilometer test track facility and 53-passenger city buses, 22 drivers were studied. b) Experiments of Forbes et al. Several experiments

4) Three Car Experiments. A series of experiments wereperformed to determine the effect on driver behavior whenthere is an opportunity for next-nearest following and offollowing the vehicle directly ahead. The degree to whicha driver uses the information that might be obtained froma vehicle two ahead was also examined. The relativespacings and the relative speeds between the first and thirdvehicles and the second and third vehicles together withthe speed and acceleration of the third vehicle wererecorded.

5) Miscellaneous Experiments. Several additional carfollowing experiments have been performed and reportedon as follows:

a) Kometani and Sasaki Experiments. Kometani andSasaki conducted and reported on a series of experimentsthat were performed to evaluate the effect of an additionalterm in the linear car following equation. This term isrelated to the acceleration of the lead vehicle. Inparticular, they investigated a model rewritten here in thefollowing form:

This equation attempts to take into account a particulardriving phenomenon, where the driver in a particular staterealizes that he should maintain a non-zero acceleration

estimators of relative acceleration. The conjecture here isthat by pursuing the task where the lead driver isundergoing a constant or near constant accelerationmaneuver, the driver becomes aware of this qualitativelyafter nullifying out relative speed - and thereby shifts theframe of reference. Such cases have been incorporatedinto models simulating the behavior of bottlenecks intunnel traffic (Helly 1959).

using three vehicle platoons were reported by Forbes et al. (1957). Here a lead vehicle was driven by one of theexperimenters while the second and third vehicles weredriven by subjects. At predetermined locations along theroadway relatively severe acceleration maneuvers wereexecuted by the lead vehicle. Photographic equipmentrecorded these events with respect to this movingreference together with speed and time. From theserecordings speeds and spacings were calculated as afunction of time. These investigators did not fit this datato car following models. However, a partial set of this datawas fitted to vehicle following models by anotherinvestigator (Helly 1959). This latter set consisted of six

��

� � ��

tests in all, four in the Lincoln Tunnel and two on an for which the correlation coefficient is a maximum and typicallyopen roadway. falls in the range of 0.85 to 0.95.

c) Ohio State Experiments. Two different sets ofexperiments have been conducted at Ohio StateUniversity. In the first set a series of subjects have been given for �, their product; C = �T, the boundary value forstudied using a car following simulator (Todosiev 1963).An integral part of the simulator is an analog computerwhich could program the lead vehicle for many differentdriving tasks. The computer could also simulate theperformance characteristics of different following vehicles.These experiments were directed toward understanding themanner in which the following vehicle behaves when thelead vehicle moves with constant speed and themeasurement of driver thresholds for changes in spacing,relative speed, and acceleration. The second set ofexperiments were conducted on a level two-lane statehighway operating at low traffic concentrations (Hankinand Rockwell 1967). In these experiments the purposewas "to develop an empirically based model of carfollowing which would predict a following car'sacceleration and change in acceleration as a function ofobserved dynamic relationships with the lead car." As inthe earlier car following experiments, spacing and relativespeed were recorded as well as speed and acceleration ofthe following vehicle.

d) Studies by Constantine and Young. These studieswere carried out using motorists in England and aphotographic system to record the data (Constantine andYoung 1967). The experiments are interesting from thevantage point that they also incorporated a secondphotographic system mounted in the following vehicle anddirected to the rear so that two sets of car following datacould be obtained simultaneously. The latter set collectedinformation on an unsuspecting motorist. Althoughaccuracy is not sufficient, such a system holds promise.

4.4.1.1 Analysis of Car Following Experiments

The analysis of recorded information from a car followingexperiment is generally made by reducing the data to numericalvalues at equal time intervals. Then, a correlation analysis iscarried out using the linear car following model to obtainestimates of the two parameters, � and T. With the data indiscrete form, the time lag ,T, also takes on discrete values. Thetime lag or response time associated with a given driver is one

The results from the preliminary experiments (Chandler et al.1958) are summarized in Table 4.1 where the estimates are

asymptotic stability; average spacing, < S >; and average speed,. The average value of the gain factor is 0.368 sec . The-1

average value of �T is close to 0.5, the asymptotic stabilityboundary limit.

Table 4.1 Results from Car-Following Experiment

Driver �� <S> ��T

1 0.74 sec 19.8 36 1.04-1

m/sec m

2 0.44 16 36.7 0.44

3 0.34 20.5 38.1 1.52

4 0.32 22.2 34.8 0.48

5 0.38 16.8 26.7 0.65

6 0.17 18.1 61.1 0.19

7 0.32 18.1 55.7 0.72

8 0.23 18.7 43.1 0.47

Using the values for � and the average spacing <S > obtained foreach subject a value of 12.1 m/sec (44.1 km/h) is obtained for anestimate of the constant a (Herman and Potts 1959). This1, 0latter estimate compares the value � for each driver with thatdriver's average spacing, <S > , since each driver is in somewhatdifferent driving state. This is illustrated in Figure 4.15. Thisapproach attempts to take into account the differences in theestimates for the gain factor � or a , obtained for different0,0drivers by attributing these differences to the differences in theirrespective average spacing. An alternate and more directapproach carries out the correlation analysis for this model usingan equation which is the discrete form of Equation 4.38 to obtaina direct estimate of the dependence of the gain factor on spacing,S(t).

��

� � ��

Figure 4.15Sensitivity Coefficient Versus the Reciprocal of the Average Vehicle Spacing (Gazis et al. 1959).

Vehicular Tunnel Experiments. Vehicular tunnels usually haveroadbeds that are limited to two lanes, one per direction.Accordingly, they consist of single-lane traffic where passing isprohibited. In order to investigate the reasonableness of the non-linear model a series of tunnel experiments were conducted.Thirty test runs in all were conducted: sixteen in the Lincoln estimate of a , and equals 29.21 km/h. Figure 4.18 graphs theTunnel, ten in the Holland Tunnel and four in the Queens Mid- gain factor, �, versus the reciprocal of the average spacing forTown Tunnel. Initially, values of the parameters for the linear the Lincoln Tunnel tests. The straight line is a "least-squares" fitmodel were obtained, i.e., � = a and T. These results are0,0

shown in Figure 4.16 where the gain factor, �= a versus the0,0

time lag, T, for all of the test runs carried out in the vehiculartunnels. The solid curve divides the domain of this twoparameter field into asymptotically stable and unstable regions.

It is of interest to note that in Figure 4.16 that many of the driversfall into the unstable region and that there are drivers who haverelatively large gain factors and time lags. Drivers withrelatively slow responses tend to compensatingly have fastmovement times and tend to apply larger brake pedal forcesresulting in larger decelerations.

Such drivers have been identified, statistically, as being involved and 14) the results tend to support the spacing dependent model.more frequently in "struck-from-behind accidents" (Babarik This time-dependent analysis has also been performed for seven1968; Brill 1972). Figures 4.17 and 4.18 graph the gain factor additional functions for the gain factor for the same fourteen

versus the reciprocal of the average vehicle spacing for the testsconducted in the Lincoln and Holland tunnels, respectively. Figure 4.17, the gain factor, �, versus the reciprocal of theaverage spacing for the Holland Tunnel tests. The straight lineis a "least-squares" fit through the origin. The slope, which is an

1 0

through the origin. These results yield characteristic speeds,a , which are within ± 3km/h for these two similar facilities.1,0

Yet these small numeric differences for the parameter al,0properly reflect known differences in the macroscopic trafficflow characteristics of these facilities.

The analysis was also performed using these test data and thenon-linear reciprocal spacing model. The results are notstrikingly different (Rothery 1968). Spacing does not changesignificantly within any one test run to provide a sensitivemeasure of the dependency of the gain factor on inter-vehicularspacing for any given driver (See Table 4.2). Where thevariation in spacings were relatively large (e.g., runs 3, 11, 13,

��

� � ��

Table 4.2Comparison of the Maximum Correlations obtained for the Linear and Reciprocal Spacing Models for the FourteenLincoln Tunnel Test Runs

Number r r <S> (m) ��S (m) Number r r <S> (m) ��S (m)0,0 l,0

1 0.686 0.459 13.4 4.2 8 0.865 0.881 19.9 3.4

2 0.878 0.843 15.5 3.9 9 0.728 0.734 7.6 1.8

3 0.77 0.778 20.6 5.9 10 0.898 0.898 10.7 2.3

4 0.793 0.748 10.6 2.9 11 0.89 0.966 26.2 6.2

5 0.831 0.862 12.3 3.9 12 0.846 0.835 18.5 1.3

6 0.72 0.709 13.5 2.1 13 0.909 0.928 18.7 8.8

7 0.64 0.678 5.5 3.2 14 0.761 0.79 46.1 17.6

0,0 l,0

Figure 4.16Gain Factor, ��, Versus the Time Lag, T, for All of the Test Runs (Rothery 1968).

��

� � ��

Note: The straight line is a "least-squares" fit through the origin. The slope, which is an estimate of a , equals 29.21 km/h.1,0

Figure 4.17Gain Factor, ��, Versus the Reciprocal of the

Average Spacing for Holland Tunnel Tests (Herman and Potts 1959).

Note: The straight line is a "least-squares" fit through the origin. The slope, is an estimate of a , equals 32.68 km/h.1,0

Figure 4.18Gain Factor, �� ,Versus the Reciprocal of the

Average Spacing for Lincoln Tunnel Tests (Herman and Potts 1959).

��

� � ��

little difference from one model to the other. There are definite of the cases when that factor is introduced and this model (�=1;trends however. If one graphs the correlation coefficient for agiven �, say �=1 versus m; 13 of the cases indicate the best fits the analysis are summarized in Figure 4.19 where the sensitivityare with m = 0 or 1. Three models tend to indicate marginal coefficient a versus the time lag, T, for the bus followingsuperiority; they are those given by (�=2; m=1), (�=1; m=0) and(�=2; m=0).

Bus Following Experiments. For each of the 22 drivers tested,the time dependent correlation analysis was carried out for thelinear model (�=0; m=0), the reciprocal spacing model (�=1;m=0), and the speed, reciprocal-spacing-squared model (�=2;m=1). Results similar to the Tunnel analysis were obtained: highcorrelations for almost all drivers and for each of the threemodels examined (Rothery et al. 1964).

The correlation analysis provided evidence for the reciprocalspacing effect with the correlation improved in about 75 percent

m=0) provided the best fit to the data. The principle results of

0,0experiments are shown. All of the data points obtained in theseresults fall in the asymptotically stable

region, whereas in the previous automobile experimentsapproximately half of the points fell into this region. In Figure4.19, the sensitivity coefficient, a , versus the time lag, T, for0,0the bus following experiments are shown. Some drivers arerepresented by more than one test. The circles are test runs bydrivers who also participated in the ten bus platoon experiments.The solid curve divides the graph into regions of asymptoticstability and instability. The dashed lines are boundaries for theregions of local stability and instability.

Table 4.3Maximum Correlation Comparison for Nine Models, a , for Fourteen Lincoln Tunnel Test Runs.

��,m

Driver r(0,0) r(1,-1) r(1,0) r(1,1) r(1,2) r(2,-1) r(2,0) r(2,1) r(2,2)

1 0.686 0.408 0.459 0.693 0.721 0.310 0.693 0.584 0.690

2 0.878 0.770 0.843 0.847 0.746 0.719 0.847 0.827 0.766

3 0.770 0.757 0.778 0.786 0.784 0.726 0.786 0.784 0.797

4 0.793 0.730 0.748 0.803 0.801 0.685 0.801 0.786 0.808

5 0.831 0.826 0.862 0.727 0.577 0.805 0.728 0.784 0.624

6 0.720 0.665 0.709 0.721 0.709 0.660 0.720 0.713 0.712

7 0.640 0.470 0.678 0.742 0.691 0.455 0.745 0.774 0.718

8 0.865 0.845 0.881 0.899 0.862 0.818 0.890 0.903 0.907

9 0.728 0.642 0.734 0.773 0.752 0.641 0.773 0.769 0.759

10 0.898 0.890 0.898 0.893 0.866 0.881 0.892 0.778 0.865

11 0.890 0.952 0.966 0.921 0.854 0.883 0.921 0.971 0.940

12 0.846 0.823 0.835 0.835 0.823 0.793 0.835 0.821 0.821

13 0.909 0.906 0.923 0.935 0.927 0.860 0.935 0.928 0.936

14 0.761 0.790 0.790 0.771 0.731 0.737 0.772 0.783 0.775

a0,0 �

a1,0

<S>

a0,0 � a2,1<S>2

��

� � ��

Note: For bus following experiments - Some drivers are represented by more than one test. The circles are test runs by driverswho also participated in the ten bus platoons experiments. The solid curve divides the graph into regions of asymptotic stabilityand instability. The dashed lines are boundaries for the regions of local stability and instability.

Figure 4.19Sensitivity Coefficient, a ,Versus the Time Lag, T (Rothery et al. 1964).0,0

The results of a limited amount of data taken in the rain suggestthat drivers operate even more stably when confronted with wetroad conditions. These results suggest that buses form a highlystable stream of traffic.

The time-independent analysis for the reciprocal-spacing modeland the speed-reciprocal-spacing-squared model uses the timedependent sensitivity coefficient result, a , the average speed,0,0, and the average spacing, <S>, for eachof the car In Figure 4.21, the sensitivity coefficient versus the ratio of thefollowing test cases in order to form estimates of a and a ,1,0 2,1i.e. by fitting

and

respectively (Rothery et al. 1964).

Figures 4.20 and 4.21 graph the values of a for all test runs0,0

versus <S> and <S> , respectively. In Figure 4.20, the-1 -2

sensitivity coefficient versus the reciprocal of the averagespacing for each bus following experiment, and the "least-squares" straight line are shown. The slope of this regression isan estimate of the reciprocal spacing sensitivity coefficient. Thesolid dots and circles are points for two different test runs.

average speed to the square of the average spacing for each busfollowing experiment and the "least-square" straight line areshown. The slope of this regression is an estimate of the speed-reciprocal spacing squared sensitivity coefficient. The solid dotsand circles are data points for two different test runs. The slopeof the straight line in each of these figures give an estimate oftheir respective sensitivity coefficient for the sample population.For the reciprocal spacing model the results indicate an estimatefor a = 52.8 ± .05 m/sec. (58 ± 1.61 km/h) and for the speed-1,0

reciprocal spacing squared model a = 54.3 ± 1.86 m. The2,1errors are one standard deviation.

��

� � ��

Note: The sensitivity coefficient versus the reciprocal of the average spacing for each bus following experiment. The least squaresstraight line is shown. The slope of this regression is an estimate of the reciprocal spacing sensitivity coefficient. The solid dots andcircles are data points for two different test runs.

Figure 4.20Sensitivity Coefficient Versus the Reciprocal of the Average Spacing (Rothery et al. 1964).

Note: The sensitivity coefficient versus the ratio of the average speed to the square of the average spacing for each bus followingexperiment. The least squares straight line is shown. The slope of this regression is an estimate of the speed-reciprocal spacingsquared sensitivity coefficient. The solid dots and circles are data points for two different test runs.

Figure 4.21Sensitivity Coefficient Versus the Ratio of the Average Speed (Rothery et al. 1964).

xn�2(t��) � �1[ �xn�1(t)� �xn�2(t)]��2[ �xn(t)� �xn�2(t)]

xn�2(t��) � �l [ �xn�1(t)� �xn�2(t)]��[ �xn(t)� �xn�2(t)]

� � �2/�1

�N

j � 1[ �x Exp.

n (j.�t) � �x Theor..n (j.�t)]2

�xnsu

�xn

��

� � ��

(4.61)

(4.62)

(4.63)

Three Car Experiments. These experiments were carried out inan effort to determine, if possible, the degree to which a driveris influenced by the vehicle two ahead, i.e., next nearestinteractions (Herman and Rothery 1965). The data collected inthese experiments are fitted to the car following model:

This equation is rewritten in the following form:

where obtained for each of the drivers for �T, exceeded the asymptotic

A linear regression analysis is then conducted for specific values contain such a term. This is not surprising, given the task ofof the parameter �. For the case � = 0 there is nearest neighborcoupling only and for � >> 1 there is next nearest

neighbor coupling only. Using eight specific values of � (0,0.25, 0.50, 1, 5, 10, 100, and �) and a mean response time of1.6 sec, a maximum correlation was obtained when � = 0.However, if response times are allowed to vary, modestimprovements can be achieved in the correlations.

While next nearest neighbor couplings cannot be ruled outentirely from this study, the results indicate that there are nosignificant changes in the parameters or the correlations when where the data has been quantitized at fixed increments of �t,following two vehicles and that the stimulus provided by thenearest neighbor vehicle, i.e., the 'lead' vehicle, is the mostsignificant. Models incorporating next nearest neighborinteractions have also been used in simulation models (Fox andLehman 1967). The influence of including such interactions insimulations are discussed in detail by those authors.

Miscellaneous Car Following Experiments. A brief discussionof the results of three additional vehicle following experimentsare included here for completeness.

The experiments of Kometani and Sasaki (1958) were carfollowing experiments where the lead vehicle's task was closelyapproximated by: "accelerate at a constant rate from a speed u toa speed u' and then decelerate at a constant rate from the speedu' to a speed u." This type of task is essentially 'closed' since the

external situation remains constant. The task does not changeappreciably from cycle to cycle. Accordingly, response timescan be reduced and even canceled because of the cyclic natureof the task.

By the driver recognizing the periodic nature of the task or thatthe motion is sustained over a period of time (� 13 sec for theacceleration phase and � 3 sec for the deceleration phase) thedriver obtains what is to him/her advanced information. Accordingly, the analysis of these experiments resulted in shortresponse times � 0.73 sec for low speed (20-40 km/h.) tests and� 0.54 sec for high speed (40-80 km/h.) tests. The results alsoproduced significantly large gain factors. All of the values

stability limit. Significantly better fits of the data can be madeusing a model which includes the acceleration of the lead vehicle(See Equation 4.58) relative to the linear model which does not

following the lead vehicle's motion as described above.

A partial set of the experiments conducted by Forbes et al.(1958) were examined by Helly (1959), who fitted test runs tothe linear vehicle model, Equation 4.41, by varying � and T tominimize the quantity:

N�t is the test run duration, (j.�t) is the experimentallyExp.

measured values for the speed of the following vehicle at timej�t, and (j.�t) is the theoretical estimate for the speed ofTheor..

the following vehicle as determined from the experimentallymeasured values of the acceleration of the following vehicle andthe speed of the lead vehicle using the linear model. Theseresults are summarized in Table 4.4.

Ohio State Simulation Studies. From a series of experimentsconducted on the Ohio State simulator, a relatively simple carfollowing model has been proposed for steady-state carfollowing (Barbosa 1961). The model is based on the conceptof driver thresholds and can be most easily described by meansof a 'typical' recording of relative speed versus spacing as

��

� � ��

Table 4.4 Results from Car Following Experiments large or the spacing becoming too small. At point "A," after a

Driver # T a r00 max

1 1.0 0.7 0.86

2 0.5 1.3 0.96

3 0.6 0.8 0.91

4 0.5 1.0 0.87

5 0.7 1.1 0.96

6 0.5 1.0 0.86

shown in Figure 4.22. At point "1," it is postulated that thedriver becomes aware that he is moving at a higher speed thanthe lead vehicle and makes the decision to decelerate in order toavoid having either the negative relative speed becoming too

time lag, the driver initiates this deceleration and reduces therelative speed to zero. Since drivers have a threshold levelbelow which relative speed cannot be estimated with accuracy,the driver continues to decelerate until he becomes aware of apositive relative speed because it either exceeds the threshold atthis spacing or because the change in spacing has exceeded itsthreshold level. At point "2," the driver makes the decision toaccelerate in order not to drift away from the lead vehicle. Thisdecision is executed at point "B" until point "3" is reach and thecycle is more or less repeated. It was found that the arcs, e.g.,AB, BC, etc. are "approximately parabolic" implying thataccelerations can be considered roughly to be constant. Theseaccelerations have been studied in detail in order to obtainestimates of relative speed thresholds and how they vary withrespect to inter-vehicle spacing and observation times (Todosiev1963). The results are summarized in Figure 4.23. This drivingtask, following a lead vehicle traveling at constant speed, wasalso studied using automobiles in a driving situation so that thepertinent data could be collected in a closer-to-reality situationand then analyzed (Rothery 1968).

Figure 4.22Relative Speed Versus Spacing (Rothery 1968).

�xn�1(t��) � �i[ �xn(t)� �xn�1(t)]

��

� � ��

(4.64)

Figure 4.23Relative Speed Thresholds Versus Inter-Vehicle Spacing for

Various Values of the Observation Time. (Rothery 1968).

The interesting element in these latter results is that the character speed stimulus is positive or negative. This effect can be takenof the motion as exhibited in Figure 4.22 is much the same. into account by rewriting our basic model as:However, the range of relative speeds at a given spacing thatwere recorded are much lower than those measured on thesimulator. Of course the perceptual worlds in these two tests areconsiderably different. The three dimensional aspects of the testtrack experiment alone might provide sufficient additional cues where � = � or � depending on whether the relative speed isto limit the subject variables in contrast to the two dimensional greater or less than zero.CRT screen presented to the 'driver' in the simulator. In anycase, thresholds estimated in driving appear to be less than those A reexamination of about forty vehicle following tests that weremeasured via simulation. carried out on test tracks and in vehicular tunnels indicates,

Asymmetry Car Following Studies. One car followingexperiment was studied segment by segment using a modelwhere the stimulus included terms proportional to deviationsfrom the mean inter-vehicle spacing, deviations from the meanspeed of the lead vehicle and deviations from the mean speed ofthe following car (Hankin and Rockwell 1967). An interestingresult of the analysis of this model is that it implied anasymmetry in the response depending on whether the relative

i + -

without exception, that such an asymmetry exists (Herman andRothery 1965). The average value of � is �10 percent greater-than � . The reason for this can partly be attributed to the fact+that vehicles have considerably different capacities to accelerateand decelerate. Further, the degree of response is likely to bedifferent for the situations where vehicles are separatingcompared to those where the spacing is decreasing. This effectcreates a special difficulty with car following models as isdiscussed in the literature (Newell 1962; Newell 1965). One of

��

� � ��

the principal difficulties is that in a cyclic change in the lead The associated units for these estimates are ft/sec, ft /sec, andvehicle's speed - accelerating up to a higher speed and then miles/car, respectively. As illustrated in this table, excellentreturning to the initial speed, the asymmetry in acceleration and agreement is obtained with the reciprocal spacing model. Howdeceleration of the following car prevents return to the original well these models fit the macroscopic data is shown in Figurespacing. With n such cycles, the spacing continues to increasethereby creating a drifting apart of the vehicles. A relaxationprocess needs to be added to the models that allows for thisasymmetry and also allows for the return to the correct spacing.

4.4.2 Macroscopic Observations: Single Lane Traffic

Several data collections on single lane traffic have been carriedout with the specific purpose of generating a large sample fromwhich accurate estimates of the macroscopic flow characteristicscould be obtained. With such a data base, direct comparisonscan be made with microscopic, car following estimates -particularly when the car following results are obtained on thesame facility as the macroscopic data is collected. One of thesedata collections was carried out in the Holland Tunnel (Edie etal. 1963). The resulting macroscopic flow data for this 24,000vehicle sample is shown in Table 4.5.

The data of Table 4.5 is also shown in graphical form, Figures4.24 and 4.25 where speed versus concentration and flow versusconcentration are shown, respectively. In Figure 4.24, speedversus vehicle concentration for data collected in the HollandTunnel is shown where each data point represents a speed classof vehicles moving with the plotted speed ± 1.61 m/sec. InFigure 4.25, flow versus vehicle concentration is shown; thesolid points are the flow values derived from the speed classesassuming steady-state conditions. (See Table 4.5 and Figure4.24.) Also included in Figure 4.25 are one-minute average flowvalues shown as encircled points. (See Edie et al. 1963). Usingthis macroscopic data set, estimates for three sensitivitycoefficients are estimated for the particular car following modelsthat appear to be of significance. These are: a , a , and a .1,0 2,1 2,0These are sometimes referred to as the Reciprocal SpacingModel, Edie's Model, and Greenshields' Model, respectively.The numerical values obtained are shown and compared with themicroscopic estimates from car following experiments for thesesame parameters.

2

4.26, where the speed versus vehicle concentration data isgraphed together with the curves corresponding to the steady-state speed-concentration relations for the various indicatedmodels. The data appears in Figure 4.24 and 4.25.

The curves are least square estimates. All three models providea good estimate of the characteristic speed (i.e., the speed atoptimum flow, namely 19, 24, and 23 mi/h for the reciprocalspacing, reciprocal spacing squared, and speed reciprocalspacing squared models, respectively).

Edie's original motivation for suggesting the reciprocal spacingspeed model was to attempt to describe low concentration, non-congested traffic. The key parameter in this model is the "meanfree speed", i.e., the vehicular stream speed as the concentrationgoes to zero. The least squares estimate from the macroscopicdata is 26.85 meters/second.

Edie also compared this model with the macroscopic data in theconcentration range from zero to 56 vehicles/kilometer; thereciprocal spacing model was used for higher concentrations(Edie 1961). Of course, the two model fit is better than any onemodel fitted over the entire range, but marginally (Rothery1968). Even though the improvement is marginal there is anapparent discontinuity in the derivative of the speed-concentration curve. This discontinuity is different than thatwhich had previously been discussed in the literature. It hadbeen suggested that there was an apparent break in the flowconcentration curve near maximum flow where the flow dropssuddenly (Edie and Foote 1958; 1960; 1961). That type ofdiscontinuity suggests that the u-k curve is discontinuous.

However, the data shown in the above figures suggest that thecurve is continuous and its derivative is not. If there is adiscontinuity in the flow concentration relation near optimumflow it is considerably smaller for the Holland Tunnel than hasbeen suggested for the Lincoln Tunnel. Nonetheless, theapparent discontinuity suggests that car following may bebimodal in character.

��

� � ��

Table 4.5 Macroscopic Flow Data

Speed Average Spacing Concentration Number of(m/sec) (m) (veh/km) Vehicles

2.1 12.3 80.1 22

2.7 12.9 76.5 58

3.3 14.6 67.6 98

3.9 15.3 64.3 125

4.5 17.1 57.6 196

5.1 17.8 55.2 293

5.7 18.8 52.6 436

6.3 19.7 50 656

6.9 20.5 48 865

7.5 22.5 43.8 1062

8.1 23.4 42 1267

8.7 25.4 38.8 1328

9.3 26.6 37 1273

9.9 27.7 35.5 1169

10.5 30 32.8 1096

11.1 32.2 30.6 1248

11.7 33.7 29.3 1280

12.3 33.8 26.8 1162

12.9` 43.2 22.8 1087

13.5 43 22.9 1252

14.1 47.4 20.8 1178

14.7 54.5 18.1 1218

15.3 56.2 17.5 1187

15.9 60.5 16.3 1135

16.5 71.5 13.8 837

17.1 75.1 13.1 569

17.7 84.7 11.6 478

18.3 77.3 12.7 291

18.9 88.4 11.1 231

19.5 100.4 9.8 169

20.1 102.7 9.6 55

20.7 120.5 8.1 56

��

� � ��

Note: Each data point represents a speed class of vehicles moving with the plotted speed ± 1 ft/sec (See Table 4.4).

Figure 4.24Speed Versus Vehicle Concentration (Edie et al. 1963).

Note: The solid points are the flow values derived from the speed classes assuming steady-state condition. Also included inFigure 4.25 are one minute average flow values shown as encircled points.

Figure 4.25Flow Versus Vehicle Concentration (Edie et al. 1963).

��

� � ��

Figure 4.26Speed Versus Vehicle Concentration (Rothery 1968).

Table 4.6 flow variables without resorting to using two differentParameter Comparison(Holland Tunnel Data)

Parameters Estimates EstimatesMicroscopic Macroscopic

a 26.8 27.81,0

a 0.57 0.122, 0

a (123) (54)2,1-1 -1

A totally different approach to modeling traffic flow variableswhich incorporates such discontinuities can be found in theliterature. Navin (1986) and Hall (1987) have suggested thatcatastrophe theory (Thom 1975; Zeeman 1977) can be used asa vehicle for representing traffic relationships. Specifically,Navin followed the two regime approach proposed by Edie andcited above and first suggested that traffic relations can berepresented using the cusp catastrophe. A serious attempt toapply such an approach to actual traffic data in order to represent

expressions or two different sets of parameters to one expressionhas been made by Hall (1987). More recently, Acha-Daza andHall (1994) have reported an analysis of freeway data usingcatastrophe theory which indicates that such an approach caneffectively be applied to traffic flow. Macroscopic data has alsobeen reported on single lane bus flow. Here platoons of tenbuses were studied (Rothery et al. 1964).

Platoons of buses were used to quantify the steady-state streamproperties and stability characteristics of single lane bus flow. Ideally, long chains of buses should be used in order to obtainthe bulk properties of the traffic stream and minimize the endeffects or eliminate this boundary effect by having the leadvehicle follow the last positioned vehicle in the platoon using acircular roadway. These latter type of experiments have beencarried out at the Road Research Laboratory in England(Wardrop 1965; Franklin 1967).

In the platoon experiments, flow rates, vehicle concentration,and speed data were obtained. The average values for the speedand concentration data for the ten bus platoon are shown inFigure 4.28 together with the numerical value for the parameter

��

� � ��

Figure 4.27Flow Versus Concentration for the Lincoln and Holland Tunnels.

Figure 4.28Average Speed Versus Concentration

for the Ten-Bus Platoon Steady-State Test Runs (Rothery 1968).

��

� � ��

a = 53 km/h which is to be compared to that obtained from the1,0two bus following experiments discussed earlier namely, 58km/h. Given these results, it is estimated that a single lane ofstandard size city buses is stable and has the capacity of over65,000 seated passengers/hour. An independent check of thisresult has been reported (Hodgkins 1963). Headway times and

speed of clusters of three or more buses on seven differenthighways distributed across the United States were measuredand concluded that a maximum flow for buses would beapproximately 1300 buses/hour and that this would occur atabout 56 km/h.

4.5 Automated Car Following

All of the discussion in this chapter has been focused on manual Mechanical Laboratory (Oshima et al. 1965), the Transportationcar following, on what drivers do in following one or more other Road Research Laboratory (Giles and Martin 1961; Cardewvehicles on a single lane of roadway. Paralleling these studies, 1970), Ford Motor Corporation (Cro and Parker 1970) and theresearch has also focused on developing controllers that would Japanese Automobile Research Institute (Ito 1973). During theautomatically mimic this task with specific target objectives in past several decades three principal research studies in this arenamind. stand out: a systems study of automated highway systems

At the 1939 World's Fair, General Motors presented program on numerous aspects of automated highways conductedconceptually such a vision of automated highways where at The Ohio State University from 1964-1980, and the Programvehicles were controlled both longitudinally (car following) and on Advanced Technology for the Highway (PATH) at thelaterally thereby freeing drivers to take on more leisurely University of California, Berkeley from about 1976 to theactivities as they moved at freeway speeds to their destinations. present. Three overviews and detailed references to milestonesIn the intervening years considerable effort has been extended of these programs can be found in the literature: Bender (1990),towards the realization of this transportation concept. One prime Fenton and Mayhan (1990), and Shladover et al. (1990),motivation for such systems is that they are envisioned to provide respectively.more efficient utilization of facilities by increasing roadwaycapacity particularly in areas where constructing additional The car following elements in these studies are focused onroadway lanes is undesirable and or impractical, and in addition, developing controllers that would replace driver behavior, carrymight improve safety levels. The concept of automated out the car following task and would satisfy one or morehighways is one where vehicles would operate on both performance and/or safety criteria. Since these studies haveconventional roads under manual control and on specially essentially been theoretical, they have by necessity required theinstrumented guideways under automatic control. Here we are difficult task of modeling vehicle dynamics. Given a controllerinterested in automatic control of the car following task. Early for the driver element and a realistic model representation ofresearch in this arena was conducted on both a theoretical and vehicle dynamics a host of design strategies and issues have beenexperimental basis and evaluated at General Motors Corporation addressed regarding inter-vehicular spacing control, platoon(Gardels 1960; Morrison 1961; Flory et al. 1962), Ohio State configurations, communication schemes, measurement andUniversity (Fenton 1968; Bender and Fenton 1969; Benton timing requirements, protocols, etc. Experimental verificationset al. 1971; Bender and Fenton 1970), Japan Governmental of these elements are underway at the present time by PATH.

conducted at General Motors from 1971-1981, a long-range

��

� � ��

4.6 Summary and Conclusions

Historically, the subject of car following has evolved over the this chapter has addressed in order. First, it provides apast forty years from conceptual ideas to mathematical model mathematical model of a relative common driving task anddescriptions, analysis, model refinements resulting from provides a scientific foundation for understanding this aspect ofempirical testing and evaluation and finally extensions into the driving task; it provides a means for analysis of local andadvanced automatic vehicular control systems. These asymptotic stability in a line of vehicles which carry implicationsdevelopments have been overlapping and interactive. There with regard to safety and traffic disruptions and other dynamicalhave been ebbs and flows in both the degree of activity and characteristics of vehicular traffic flow; it provides a steady stateprogress made by numerous researchers that have been involved description of single lane traffic flow with associated roadin the contributions made to date. capacity estimates; it provides a stepping stone for extension into

The overall importance of the development of the subject of car will undoubtedly continue to provide stimulus andfollowing can be viewed from five vantage points, four of which encouragement to scientists working in related areas of traffic

advance automatic vehicle control systems; and finally, it has and

theory.

References

Acha-Daza, J. A. and F. L. Hall (1994). Application ofCatastrophe Theory to Traffic Flow Variables.Transportation Research - B, 28B(3). Elsevier ScienceLtd., pp. 235-250. Constantine, T. and A. P. Young (1967). Traffic Dynamics:

Babarik, P. (1968). Automobile Accidents and Driver Car Following Studies. Traffic Engineering and Control 8,Reaction Pattern. Journal of Applied Psychology, 52(1),pp. 49-54. Cumming, R. W. (1963). The Analysis of Skills in Driving.

Barbosa, L. (1961). Studies on Traffic Flow Models. ReportsNo. 202A-1. The Ohio State University Antenna Darroch, J. N. and R. W. Rothery (1973). Car Following andLaboratory.

Bender, J. G. (1971). An Experimental Study of Vehicle Symposium on the Theory of Traffic Flow andAutomatic Longitudinal Control. IEEE Transactions onVehicular Technology, VT-20, pp. 114-123.

Bender, J. G. (1991). An Overview of Systems Studies ofAutomated Highway Systems. IEEE Transactions onVehicular Technology 40(1). IEEE Vehicular TechnologySociety, pp. 82-99. Drew, D. R. (1965). Deterministic Aspects of Freeway

Bender, J. G. and R. E. Fenton (1969). A Study of Automatic Operations and Control. Highway Research Record, 99,Car Following. IEEE Transactions on VehicularTechnology, VT-18, pp. 134-140. Edie, L. C. (1961). Car-Following and Steady State Theory

Cardew, K. H. F. (1970). The Automatic Steering of Vehicles for Non-Congested Traffic. Operations Research 9(1),-An Experimental System Fitted to a Citroen Car.RL 340, Road Research Laboratories. Edie, L. C. and E. Baverez (1967). Generation and

Chandler, F. E., R. Herman, and E. W. Montroll, (1958). Traffic Dynamics: Studies in Car Following, Operations Science Proceedings of the 3rd International Symposium onResearch, 6, pp. 165-184. the Theory of Traffic Flow. L.C. Edie, R. Herman and R.W.

Chow, T. S. (1958). Operational Analysis of a TrafficDynamics Problem. Operations Research, 6(6),pp. 165-184.

pp. 551.

Journal of the Australian Road Research Board 1, pp. 4.

Spectral Analysis. Proceedings of the 5th International

Transportation. Ed. Newell, G. F., American ElsevierPublishing Co., New York.

Drake, J. S., J. L. Schofer, and A. D. May, Jr. (1967). AStatistical Analysis of Speed Density Hypotheses. HighwayResearch Record 154, pp. 53-87.

pp. 48-58.

pp. 66-76.

Propagation of Stop-Start Waves. Vehicular Traffic

Rothery (Eds.). American Elsevier, New York.

��

� � ��

Edie, L. C. and R. S. Foote, (1958). Traffic Flow in Tunnels. Greenburg, H. (1959). An Analysis of Traffic Flow.Proceedings of the Highway Research Board, 37, Operations Research 7(I), pp. 255-275.pp. 334-344.

Edie, L. C. and R. S. Foote (1960). Effect of Shock Waves on Proceedings of the Highway Research Board, 14, 468.Tunnel Traffic Flow. Proceedings of the Highway Research Hall, F. L. (1987). An Interpretation of Speed-FlowBoard, 39, pp. 492-505.

Edie, L. C., and R. S. Foote (1961). Experiments on Single Transportation Research, 21A, pp. 335-344.Lane Flow in Tunnels. Theory of Traffic Flow Proceedings Hankin, A. and T. H. Rockwell (1967). A Model of Carof the Theory of Traffic Flow. Ed. R. Herman. ElsevierPub. Co., Amsterdam, pp. 175-192.

Edie, L. C., R. S. Foote, R. Herman, and R. W. Rothery(1963). Analysis of Single Lane Traffic Flow. TrafficEngineering, 21, 27.

Ellson, D. G. (1949). The Application of Operational Analysisto Human Motor Behavior. Psychology Review 9,pp. 56.

Fenton, R. E. (1968). One Approach to Highway Automation.Proceedings IEEE 56; pp. 556-566.

Fenton, R. E. and R. J. Mayhan (1991). Automated HighwayStudies at The Ohio State University - An Overview. IEEETransaction on Vehicular Technology 40(1); IEEEVehicular Technology Society, pp. 100-113.

Flory, L. E. (1962). Electronic Techniques in a System ofHighway Vehicle Control. RCA Review 23, pp. 293-310.

Forbes, T. W., M. J. Zagorski, E. L. Holshouser, and W. A.Deterline (1959). Measurement of Driver Reaction toTunnel Conditions. Proceedings of the Highway ResearchBoard 37, pp. 345-357.

Gardels, K. (1960). Automatic Car Controls for ElectronicHighways. GM Research Laboratories Report GMR-276,Warren, MI.

Gazis, D. C., R. Herman, and R. B. Potts (1959). CarFollowing Theory of Steady State Traffic Flow. OperationsResearch 7(4), pp. 499-505.

Gazis, D. C., R. Herman, and R. W. Rothery (1961). Non-Linear Follow the Leader Models of Traffic Flow.Operations Research, 9, pp. 545-567.

Gazis, D. C., R. Herman, and R. W. Rothery (1963). Analytical Methods in Transportation: Mathematical Car-Following Bonn, Germany.Theory of Traffic Flow. Journal of the EngineeringMechanics Division, ASCE Proc. Paper 3724 89(Paper372), pp. 29-46. Hodgkins, E. A. (1963). Effect of Buses on Freeway Capacity.

Giles, C. G. and J. A. Martin (1961). Cable Installation forVehicle Guidance Investigation in the New Research Trackat Crowthorne. Rep RN/4057/CGG, Road ResearchLaboratory, Crowthorne, England.

Greenshields, B. D. (1935). A Study of Traffic Capacity.

Concentration Relationships Using Catastrophe Theory.

Following Derived Empirically by Piece-Wise RegressionAnalysis. Vehicular Traffic Science Proceedings of the 3rdInternational Symposium on the Theory of Traffic Flow.L.C. Edie, R. Herman and R.W. Rothery (Eds.). AmericanElsevier, New York.

Harris, A. J. (1964). Following Distances, Braking Capacityand the Probability of Danger of Collision BetweenVehicles. Australian Road Research Board, Proceedings 2,Part 1, pp. 496-412.

Helly, W. (1959). Dynamics of Single-Lane Vehicular TrafficFlow. Research Report No. 2, Center for OperationsResearch, MIT. Cambridge, Mass.

Herman, R., E. W. Montroll, R. B. Potts and R. W. Rothery(1958). Traffic Dynamics: Analysis of Stability in CarFollowing. Operations Research, E. 17, pp. 86-106.

Herman, R. and R. B. Potts (1959). Single Lane TrafficTheory and Experiment. Proceedings Symposium onTheory of Traffic Flow. Ed. R. Herman, ElsevierPublications Co., pp. 120-146.

Herman, R. and R. W. Rothery (1962). Microscopic andMacroscopic Aspects of Single Lane Traffic Flow.Operations Research, Japan, pp. 74.

Herman, R. and R. W. Rothery (1965). Car Following andSteady-State Flow. Proceedings of the 2nd InternationalSymposium on the Theory of Traffic Flow. Ed J. Almond,O.E.C.D., Paris.

Herman, R. and R. W. Rothery (1969). Frequency andAmplitude Dependence of Disturbances in a TrafficStream. Proceedings of 4th International Symposium on theTheory of Traffic Flow, Ed. W. Leutzbach and P. Baron.

Highway Capacity Manual (1950). U.S. Government PrintingOffice, Washington, DC.

Presentation at the 43rd Annual Meeting of the HighwayResearch Board 33. Highway Research Abstracts, pp. 69.

Ito, T. (1973). An Automatic Drivers System of Automobilesby Guidance Cables. Society of Automotive Engineering(730127).

��

� � ��

Kometani, E. and T. Suzaki (1958). On the Stability of TrafficFlow. J. Operations Research, Japan 2, pp. 11-26.

Lam, T. and R. W. Rothery (1970). Spectral Analysis ofSpeed fluctuations on a Freeway. Transportation Science4(3).

Lee, G. (1966). A Generalization of Linear Car-FollowingTheory. Operations Research 14(4), pp. 595-606.

May, A. D. and H.E.M. Keller (1967). Non-Integer CarFollowing Models. Highway Research Record 199, pp. 19-32.

Michaels, R. M. (1963). Perceptual Factors in CarFollowing. Proceedings of the 2nd InternationalSymposium on the Theory of Road Traffic Flow (London,England), OECD.

Navin, F. P. D. (1986). Traffic Congestion Catastrophes. Transportation Planning and Technology 11, pp. 19-25.

Newell, G. F. (1961). Nonlinear Effects in the Dynamics ofCar Following. Operations Research 9(2), pp. 209-229.

Newell, G. F. (1962). Theories of Instability in DenseHighway Traffic. J. Operations Research Society of Japan5(1), pp. 9-54.

Oshima, R. (1965). Control System for Automobile Driving.Proceedings of the Tokyo 1FAC Symposium, pp. 347-357.

Pipes, L. A. (1951). A Proposed Dynamic Analogy of Traffic.ITTE Report, Institute of Transportation and TrafficEngineering, University of California, Berkeley.

Pipes, L. A. (1953). An Operational Analysis of TrafficDynamics. Journal of Applied Physics 24, pp. 271-281.

Reuschel, A. (1950). Fahrzeugbewegungen in der KolonneBeigleichformig beschleunigtem oder vertzogertenLeitfahrzeub, Zeit. D. Oster. Ing. U. Architekt Vereines Ed.(Vehicle Movements in a Platoon with UniformAcceleration or Deceleration of the Lead Vehicle), pp. 50-62 and 73-77.

Rothery, R. W., R. Silver, R. Herman and C. Torner (1964). Analysis of Experiments on Single Lane Bus Flow.Operations Research 12, pp. 913.

Shladover, S. E., C. A. Desoer, J. D. Hedrick, M. Tomizuka,J. Walrand, W. B. Zhang, D. H. McMahon, H. Peng, S. Shiekholeslam, and N. McKeown (1991). AutomatedVehicle Control Developments in the PATH Program.IEEE Transactions on Vehicular Technology 40(1), pp. 114-130.

Taylor, F. V. (1949). Certain Characteristics of the HumanServe. Electrical Engineering 68, pp. 235.

Telfor, G. W. (1931). The Refractory Phase of Voluntary andAssociative Responses. Journal of ExperimentalPsychology, 14, pp. 1.

Thorn, R. (1975). Structural Stability and Morphogenesis.An English Translation by D.H. Fowler. John Wiley &Sons, New York.

Todosiev, E. P. (1963). The Action Point Model of the DriverVehicle System. Report No. 202A-3. Ohio StateUniversity, Engineering Experiment Station, Columbus,Ohio.

Tuck, E. (1961). Stability of Following in Two Dimensions.Operations Research, 9(4), pp. 479-495.

Tustin, A. (1947). The Nature of the Operator Response inManual Control and its Implication for Controller Design.J.I.E.E. 92, pp. 56.

Unwin, E. A. and L. Duckstein (1967). Stability of Reciprocal-Spacing Type Car Following Models.Transportation Science, 1, pp. 95-108.

Uttley, A. (1941). The Human Operator as an IntermittentServo. Report of the 5th Meeting of Manual TrackingPanel, GF/171.SR1A.

Wardrop, J. G. (1965). Experimental Speed/Flow Relations ina Single Lane. Proceedings of the 2nd InternationalSymposium on the Theory of Road Traffic Flow. Ed. J.Almond O.E.C.D.

Zeeman, E. C. (1977). Catastrophe Theory. Selected Papers1972-1977. Addison-Wesley, London.

CONTINUUM FLOW MODELS

REVISED BY H. MICHAEL ZHANG* ORIGINAL TEXT BY REINHART KUHNE7

PANOS MICHALOPOULOS8

* Professor, Department of Civil and Environmental Engineering, University of California Davis, One Shields Avenue, Davis, CA 95616

7 Managing Director, Steierwald Schonharting und Partner, Hessbrühlstr. 21c 7 70565 Stuttgart, Germany

8 Professor, Department of Civil Engineering, University of Minnesota Institute of Technology, 122 Civil Engineering Building, 500 Pillsbury Drive S.E., Minneapolis, MN 55455-0220�

c 20

g nj

g nj

k nj ,q n

j

ue(kn

ij)

�x


a = dimensionless traffic parameter L = distanceA = stop-start wave amplitude L = length of periodic interval� = sensitivity coefficientb = net queue length at traffic signalc = g + r = cycle length � = wave length of stop-start wavesc = coefficient0

= constant, independent of density kds = infinitesimal time�t, �x = the time and space increments respectively

such that �x/�t > free flow speed �� , �� = deviationsi i+1� = state vector� , � = state vector at position i, i+1i i+1

f(x, v, t ) = vehicular speed distribution functionf = relative truck portion, k = kpassf = equilibrium speed distribution0� = fluctuating force as a stochastic quantity q k = arrival flow and density conditionsg = effective green interval q = capacity flow

= is the generation (dissipation) rate at node j at r = effective red intervalt = t + n�t; if no sinks or sources exist =00 and the last term of Equation 5.28 vanishes T = oscillation time

g = minimum green time required for t = timeminundersaturation

h = average space headway � = relaxation time as interaction time lagi = station u = speedj = node U (k) = equilibrium speed-density relationk = densityk , k = density downstream, upstream shock- +k = operating point0

k = equilibrium density10

K = constant valueA

k = density within La 2

k = density "bumper to bumper"bumperk , q = density, flow downstreamd dk , q = density, flow upstreamu uk = vehicle density in homogeneous flowhomk = jam density of the approach underj

consideration = density and flow rate on node j at

t = t + n�t0

k = density conditions mk = density "bumper to bumper" for 100%pass

passenger cars y(t) = queue length at any time point tk = reference state y = queue length from i to j assuming a positiverefk = density "bumper to bumper" for 100% trucks direction opposite to x, i.e. from B to Atruck

ld = logarithmus dualisl = characteristic lengtho

μ = dynamic viscosity0μ = viscosity termn = current time stepN = normalization constantn , n = exponentsi 2

N = number of cars (volume)i� = eigenvaluep = probabilityq = actual traffic volume, flowQ = net flow rate0q = average flow ratea

a a

ni

= quantity0

t = the initial time0

e= equilibrium speed

u = free-flow speed of the approach underfconsideration

u = group velocitygu - u = speed rangemax minu = shock wave speedwu = spatial derivative of profile speedzv(k) = viscosityv = values of the group velocitygW(q) = distribution of the actual traffic volume

values qx = spacexh = estimated queue lengthx , t , y = coordinates at point ii i iX = length of any line ijijy = street width

ij

z = x - U, t, collective coordinate= shockspeed

Revised Chapter 5 of Traffic Flow Theory Monograph

Original text by Reinhart Kuhne and Panos MichalopoulosRevised by H. M. Zhang

1 Conservation and traffic waves

There have been two kinds of traffic flow theories presented thus far in this monograph. Chapter2 discussed the relations among aggregated quantities of flow rate, concentration and mean speedsfor stationary vehicular traffic (these relations are also referred to as the equations of state intraffic flow), and Chapter 3 described the dynamic evolution of microscopic quantities of vehiclespacing, headway and speeds in single-lane traffic through the follow-the-leader type of models.In this Chapter we study another kind of traffic flow theory—continuum traffic flow theory—that describes the temporal-spatial evolution of macroscopic flow quantities as mentioned earlier.Continuum theories are natural extensions to the first two kinds of theories because, one onehand, they are closely related to the equations of state and car-following theories, and on theother hand, they overcome some of the drawbacks of the first (stationarity) and second (too muchdetail and limited observability) kinds of theories. The quantities and features that continuumtheories describe, such as traffic concentration and shock waves, are mostly observable with currentsurveillance technology, which makes it easier to validate and calibrate these models.

Many theories exist in the continuum description of traffic flow. All of them share two funda-mental relations: one is the conservation of vehicles and the other is the flow-concentration-speedrelation

q = ku. (1)

The q − k − u relation is true by choice, i.e., one defines flow rate (q), concentration (k) and meanspeed (u) in such a way that (1) always holds (see Chapter 2 for definitions). Conservation ofvehicles, on the other hand, is true regardless of how q, k, u are defined, and can be expressed indifferent forms. Here we present three forms of the conservation law using variables in (1).

The first integral form of the conservation law. Consider a stretch of highway between x1 andx2 (x1 < x2). At time t the traffic concentration on this section is k(x, t), and traffic flows intothe section at a rate of q(x1, t), and out of the section at a rate of q(x2, t). The total number of

5-1

vehicles in this section at time t is then ∫ x2

x1

k(x, t)dx.

Suppose no entries and exits exist between x1 and x2, then by conservation the rate of change inthe number of vehicles

∂

∂t

∫ x2

x1

k(x, t)dx.

should equal to the net flow into this section

q(x1, t) − q(x2, t),

that is,∂

∂t

∫ x2

x1

k(x, t)dx = q(x1, t)− q(x2, t), (2)

This is the first integral form of the conservation law.The second integral form of the conservation law. The conservation of vehicles also means that,

in the x−t plane, the net number of vehicles passing through any closed curve is zero provided thatno sources and sinks are there in the enclosed region (Fig. 2). Note that the number of vehiclesacross any segment dl of curve C is −kdx + qdt, the total number of vehicles across the enclosedregion is therefore the line integral of the vector field (−k, q), and we have the second integral formof the conservation law: ∫

C−kdx + qdt = 0. (3)

(3) implies that we can construct a scalar function N(x, t) with the properties

q =∂N

∂t, k = −∂N

∂x.

This function is the cumulative count of vehicles passing location x at time t, provided that N(x, t =0) = 0. The fact that such a function exists is another consequence of the conservation law.

The differential form of the conservation law. By applying the divergence theorem to the secondintegral form we obtain ∫

C−kdx + qdt =

∫ ∫D

(∂k

∂t+

∂q

∂x

)dxdt = 0,

which leads to the differential form of the conservation law:

∂k

∂t+

∂q

∂x= 0. (4)

The three different forms of the conservation law are not completely equivalent. Note that inthe differential form, both k and q are required to be differentiable with respect to time t and spacex. In contrast, it is perfectly fine in the integral forms that these variables are discontinuous inspace and/or time. This is an important distinction because a special kind of traffic waves, calledshock waves, are prevalent in vehicular traffic flow and play an essential role in the continuum

5-2


description of traffic flow. A shock wave is a drastic change in traffic concentration (and/or speed,flow rate) that propagates through a traffic stream. Examples of shock waves include the stoppageof traffic in front of a red light, and traffic slow-downs caused by an accident. In reality a shockalways has a profile (that is, a transition region of non-zero width) that usually spans a few vehiclelengths, but this can be practically treated as a discontinuity when compared with the length ofroads in consideration.

Because of the existence of shocks, it is necessary to expand the solution space of (4) to includethe so-called weak solutions. In mathematical sense a weak solution is a function (k, q)(x, t) thatsatisfies (4) everywhere except along a certain path x(t). On x(t) (k, q)(x, t) are discontinuous butobey the integral forms of the conservation law. A shock wave solution to (4) is a weak solution.Moreover, its path x(t) is governed by the following equation:

x(t) = [q]/[k]. (5)

where [k] = kr − kl, [q] = qr − ql, x(t) is the speed of the shock, also denoted as s, and (kl,r, ql,r) arethe traffic states immediately to the right and left of the shock path x(t), respectively. Expression(5) is also known as the Rankine-Hugoniot (R-H) condition (LeVeque 1992) , which is a consequenceof the conservation law.

Expression (5) has a simple geometric interpretation in the (k, q) plane, also known as the k− q

phase plane. It is simply the slope of the segment that connects the two phase points (kl, ql) and(kr, qr) (Fig. 1(a)). Suppose that a unique curve connects (kl,r, ql,r), and kr → kl (i.e., the shockis weak), then s approaches the slope of the tangent of the connecting curve at (kl, ql) (Fig. 1(b)).We call this speed the speed of traffic sound waves, and give it a special notation c(k). This is thespeed that information is propagated in homogeneous traffic. It plays a central role in the basickinematic wave model of traffic flow. It must be pointed out that a unique q − k curve is notnecessary to compute the shock speed. This can be done as long as both the densities and flowrates on both sides of the shock are known.

Next we provide two derivations of (5) from the conservation law. The first derivation is basedon the first integral form of the conservation law. Suppose that traffic states on both sides of theshock are constant states (k1,2, q1,2), and suppose that two observers located at x1(t) and x2(t) onthe two sides of the shock (i.e., x1 < x(t) < x2) travel at precisely the speed of the shock, s. Weapply the conservation law (2) to the region between the two observers after doing a coordinatetransformation x

′= x − st to obtain:

∂

∂t

∫ x′2

x′1

k(x, t)dx = [q1 − k1s]− [q2 − k2s].

The integral, however, can be evaluated independently in [x′1, x

′l) and (x

′r , x

′2], where x

′l,r are points

immediate to the right/left of shock path x′(t). This integral turns out to be

k1(x′l − x

′1) + k2(x

′2 − x

′r),

5-3

Figure 1: Geometric representation of shocks, sound waves, and traffic speeds in the k − q phaseplane

a constant, whose derivative is zero. Therefore we have

[q1 − k1s] − [q2 − k2s] = 0,

which leads to (5) after rearranging terms.The second derivation of the Rankine-Hugoniot condition comes from the second integral form.

As shown in Fig.2, a shock path x(t) breaks the closed curve C into two parts Cl and Cr. Let Γl,r

be the right/left edges of x(t), respectively, then the conservation law applies to each of the threeenclosures C, Cl ∪ Γl, and Cr ∪ Γr: ∫

C−kdx + qdt = 0. (6)∫

Cl∪Γl

−kdx + qdt = 0. (7)∫Cr∪Γr

−kdx + qdt = 0. (8)

Adding (7) and (8) together and rearranging terms we obtain∫C−kdx + qdt +

∫Γl

−kdx + qdt +∫Γr

−kdx + qdt = 0.

The first integral is zero, and the second and third integrand become

(−kl,rs + ql,r)dt

on Γl,r. Taking account the direction of integration we obtain the following integral equation:∫ t2

t1{(−kls + ql)− (−krs + qr)}dt = 0.

5-4


This implies that(−kls + ql)− (−krs + qr) = 0,

which also leads to the Rankine-Hugoniot shock condition.

Figure 2: Field representation of shocks and conservation of flow

The differential conservation law (4), supplemented by the Rankine-Hugoniot condition (5), isthe corner stone of all continuum vehicular traffic theories. Yet it is not a complete theory, nor is itunique to traffic flow. In fact, the differential conservation law is obeyed by many kinds of materialfluids, such as gas flow in a pipe or water flow in a channel. As such, it does not capture the uniquecharacters of vehicular traffic flow, a special kind of “fluid”. It must be supplemented by additionalrelations to form a complete and useful traffic flow theory. Some of the relations between macro-scopic traffic variables are already available and have been discussed in Chapter 2. These are thebinary relations between flow rate, traffic concentration and mean travel speed. Although these re-lations are found among stationary traffic, they can also be used as a first approximation of dynamicflow-concentration-speed relations. This leads to the first, and the simplest continuum theory oftraffic flow—the kinematic wave theory developed independently by Lighthill and Whitham (1955),and Richards (1956) (The LWR model). A more accurate relation between flow-concentration orspeed-concentration is one that accounts for the dynamic behavior of drivers—anticipation andinertia. The usage of such a relation leads to the so-called higher-order models. We will study inthis chapter these two kinds of continuum theories in great deal. In the sections to follow, we’llfirst study the properties of the LWR model and its Riemann problem, then study the propertiesof a special class of higher-order models and their respective Riemann problems. Next we developnumerical approximations of both types of models, and finally we provide some examples of theuse of the numerical approximations.

5-5


2 The kinematic wave model of LWR

2.1 The LWR model and characteristics

The LWR model assumes that the relation observed in stationary flow, q = f∗(k), also applies todynamic traffic. With such a relation, the conservation law becomes:

kt + f∗(k)x = 0. (9)

and the Rankine-Hugoniot condition becomes

s = [f∗]/[k].

This is the celebrated kinematic wave model of LWR. It is one of the simplest nonlinear scalarconservation laws in physics and engineering.

Experimental evidence indicates that f∗(k) is usually smooth and concave (refer to Chapter 2),and satisfies the following boundary conditions:

f∗(0) = f∗(kj) = 0,

f′∗(0) = vf > 0, f∗′(kj) = cj < 0.

where kj is the jam concentration at which vehicles grind to a halt, usually ranging from 260 vpmto 330 vpm (in passenger car units), cj is the speed of traffic sound waves at jam condition, takingvalues in the range of [-10 mph, -20 mph], and vf is free-flow travel speed, ranging from 55mph

to 75mph on freeways. All these parameter values are readily computable from conventional fieldmeasurements.

With the introduction of traffic wave speed ( when no confusion arises we use traffic wave andtraffic sound waves interchangably) c(k) = f∗′(k), the LWR model also reads

kt + c(k)kx = 0. (10)

Now let us look at what changes in concentration an observer sees when he travels at the speed ofc(k), that of traffic waves.

dk

dt= kt + kx

dx

dt= kt + c(k)kx = 0.

That is to say, he sees no changes in density at all if he travels at the wave speed. As a result, if heknows the initial concentration at a point ξ, he’ll know the concentration at any point on the path

x(t) = c(k), x(0) = ξ. (11)

This path is called the characteristic curve of the LWR model, and the wave speed c(k) is alsoknown as the characteristic speed of the LWR model. Because k is constant along the characteristiccurve, the characteristic of (9) is therefore a straight line.

One must not confuse over traffic sound wave speeds, shock wave speeds, characteristic speeds,and vehicle speeds. Traffic sound wave speeds and characteristic speeds in the LWR model are

5-6

identical, which are the slopes of the tangential lines on the flow-concentration curve (also knownas the fundamental diagram of traffic flow). Shock wave speeds are the slopes of the secant thatconnecting any two traffic states on f∗(k) , and vehicular speeds are the slopes of the rays from theorigin to points on f∗(k) (Fig. 1). Because of the concave shape of f∗(k), we have the followingrelations among the various speeds:

v = f∗/k ≡ v∗(k) ≥ c(k), s < v∗(kl,r),

that is to say, all waves, including sound and shock waves, travel no faster than traffic. In otherwords, information in LWR traffic is propagated against the traffic stream. This property ofinformation propagation in the LWR model is known as the anisotropic property of traffic flow.The anisotropic property may not hold if the fundamental diagram is not concave (Zhang 2000c).

2.2 The Riemann problem and entropy solutions

The LWR model is a well-posed hyperbolic partial differential equation (pde) 1 and can be solvedwith proper initial/boundary data. In fact, analytical solutions can be obtained for a special kindof problem called Riemann problem, whose initial data—the so-called Reimann data—are twoconstant states kl,r ≥ 0 separated by a single jump:

k(x, t = 0) =

{kl, x < 0,

kr, x > 0

The analytical solution to the Riemann problem can either be a shock:

k(x, t) =

{kl, x < st,

kr , x > st.(12)

or a smooth expansion wave (also known as a rarefaction wave):

k(x, t) =

⎧⎪⎪⎨⎪⎪⎩

kl, x < clt,

(f′∗)−1

(xt

), clt ≤ x ≤ crt,

kr, x > crt

(13)

The condition for it to be a shock is governed by the so-called entropy condition, which states that

cl > cr. (14)

Otherwise the solution will be an expansion wave. Because of the concavity condition f′′∗ (k) < 0,

(14) implies that kl < kr , that is, shocks arising from the LWR theory are compressive. Moreover,they reach vehicles from upstream because s < vl,r. Examples of the two kinds of solutions to aRiemann problem are shown in Figs. 3& 4.

For general initial conditions, it is usually tedious, if not difficult, to obtain analytical solutionsof the LWR equation, and numerical approximations are often sought, where the Riemann problem

1A pde is hyperbolic if its characteristic speeds are real (as compared to imaginary).

5-7

Figure 3: A shock solution

Figure 4: A rarefaction solution

plays a key role in developing some of the most efficient and accurate numerical schemes. This willbe discussed in Section 4 together with the treatment of boundary conditions. There are specialcases where an analytical solution is still straightforward to obtain. One case involves a special kindof initial condition—k(x, 0) is piece-wise constant and and increasing, and another case involves aspecial kind of fundamental diagram—the triangular shaped fundamental diagram that has onlytwo wave speeds (Koshi et al 1983, Newell 1993). In the former case, a series of Riemann problemscan be solved with consideration of wave interactions, and in the latter case, expansion waves arereplaced by acceleration shocks, and it is much simpler to construct the solutions with shocks thanexpansion waves (see Newell 1993).

2.3 Applications

Same as Sections 5.1.3 & 5.1.4 in the original text.

5-8


2.4 Extensions to the LWR model

The LWR model we presented applies to traffic on a homogeneous highway with no entering andexiting traffic. In reality all highways have entries and exits and variable geometric features.Fortunately the LWR equation can be easily extended to model such inhomogeneities.

Suppose the net inflow to a road section is s(x, t)dxdt from sources such as freeway ramps, thenfrom conservation we have

kt + f∗(k)x = r(x, t), (15)

This is the LWR model with sources. These source flows can be entering or exiting flows fromramps, or both.

When road geometries change, such as the drop of a lane at some locations, they often generatedisturbances that cause traffic breakdowns. The effect of such inhomogeneities on traffic flow can becaptured to some degree with a location-dependent fundamental diagram, f∗(k, x). Because vehicleconservation still holds, applying the conservation principle to an inhomogeneous road leads to thefollowing extended LWR model:

kt + f∗(k, x)x = 0. (16)

If a road segment has both sources and geometric inhomogeneities, the following model applies:

kt + f∗(k, x)x = r(x, t). (17)

Like the homogeneous LWR model, these extensions are also non-linear hyperbolic pdes, whosecharacteristics are

x = f∗k(k, x). (18)

These characteristics, however, are no longer straight lines as in the homogeneous model. This isbecause along the characteristics, we have

k = kt + kxx = kt + f∗kkx =

⎧⎪⎪⎨⎪⎪⎩

r, for (15)−f∗x, for (16)

r − f∗x, for (17)(19)

(18) and (19) form a system of ordinary differential equations (ode), and can be solved iterativelyusing any ode solver such as the Runge-Kutta method. This way of solving the inhomogeneousLWR models is known as the method of characteristics (Courant and Hilbert 1962).

The basic LWR model and its various extensions we have covered treat traffic across lanes ashomogeneous, which is somewhat restrictive because alternate motion of traffic in different lanes isoften observed in heavily congested traffic. This limitation is easily overcome by modeling trafficevolution in each lane. Here the conservation principle still applies, but a continuum of sourcesexist for each lane. These sources are the exchange of flow between adjacent lanes. Without lossof generality, assume a two-lane highway with traffic (k1(x, t), q1(x, t)) and (k2(x, t), q2(x, t)). LetQ1 be the net amount of traffic entering lane 1 from lane 2 in unit time and distance, and Q1 be

5-9


the net amount of traffic entering lane 2 from lane 1 in unit time and distance, then the extendedLWR model for a two-lane highway is (Dressler 1949)

k1t + q1x = Q1, (20)

k2t + q2x = Q2. (21)

where q1 = f1∗(k1) and q2 = f2∗(k2).From conservation we know the source flows Q1, Q2 observe

Q1 + Q2 = 0

and experiences tell us that the change of flow between lanes is related to the differential of lanespeeds/densities, for drivers always like to move into the faster lane in congested traffic. An exampleof Q1, Q2 was proposed by Gazis et al. (1962):

Q1,2 = α[(k2,1 − k1,2) − (k02,1 − k0

1,2)] (22)

where k01,2 are equilibrium densities of lanes 1 & 2, respectively (Gazis et al. 1962). These can be

observed experimentally.The above lane-specific LWR model is a system of quasi-linear PDEs and can be expressed in

vector forms (k1

k2

)t

+

(f1∗(k1)f2∗(k2)

)x

=

(Q1(k1, k2)Q2(k1, k2)

), (23)

or (k1

k2

)t

+

(f1∗,k1 0

0 f2∗,k2

)(k1

k2

)x

=

(Q1(k1, k2)Q2(k1, k2)

). (24)

For a system of quasi-linear PDEs, the characteristics are the eigenvalues of the Jacobian ma-

trix DF ≡(

f1∗,k1 00 f2∗,k2

), which contains the partial derivatives of the flow vector F (U) ≡

(f1∗(k1)f2∗(k2)

)with respect to the state vector U ≡

(k1

k2

).

This system is said to be strictly hyperbolic if all its characteristics are real and distinct. Itis easy to see that the characteristics of (23) are f

′1∗(k1) and f

′2∗(k2) because the dynamics of the

two lanes are nearly decoupled (the only coupling comes from the source term). The system isstrictly hyperbolic under the condition that f1∗(k1) �= f2∗(k2) and f

′′1,2∗ < 0. Thus this system can

be solved in the same way as other inhomogeneous models through the method of characteristics.Only this time one solves a system of four odes instead of two:

x1 = f′1∗(k1),

x2 = f′2∗(k2),

k1 = Q1(k1, k2),

k2 = Q2(k1, k2).

It should be noted that both the inhomogeneous models and the multiple-lane models can besolved numerically using the finite difference procedures developed in Section 5.

5-10

2.5 Limitations of the LWR model

The kinematic wave model of LWR is only a first approximation of the real traffic flow process andis deficient in describing a number of traffic features of potential importance. The include (i) driverdifferences, (ii) shock structure, (iii) forward moving waves in queued up traffic and (iv) trafficinstability (Daganzo 1997).

The LWR model is inadequate for modeling light traffic because it does not recognize thesegregation of fast and slow drivers in the traffic stream owing to passing. An adequate model fortraffic where significant amount of passing takes place should track both fast and slow drivers andtheir interactions, not by lumping both types of drivers together through a single descriptor such asdensity k(x, t). When drivers travel at roughly the same speed, however, the LWR model can stillbe used for light traffic, where density is independent of concentration and flow increases linearlywith density.

The second deficiency of the LWR model lies in its treatment of shock waves. The shocks inthe LWR model has no width, i.e., no transition region. A vehicle enters a shock thus dropping itsspeed in no time, implying an infinite deceleration. In reality, shocks always have a structure, i.e., atransition region of a few vehicles’ length in which vehicles decelerate at finite rates. This deficiencycan be addressed through introducing higher-order approximations of traffic dynamics (Kuhne1984,1989) or using a microscopic description (Newell 1961). It is one of the author’s (Zhang)opinion that as long as the accurate knowledge of vehicle acceleration is not of main interest, thelack of a shock structure is not a major failing of the LWR theory, because in comparison withthe space scale one is modeling, a few vehicle lengths of shock width can be practically consideredas nil. To most applications, the important matter is whether the LWR theory gives a reasonableestimate of shock speeds. Empirical evidence indicates that it does (Chapter 4 in Daganzo 1997).

The LWR theory has only one family of waves which travels at a speed of f′∗(k). These waves

always travel against the traffic stream because f′∗(k) < v∗(k). In particular, f

′∗(k) < 0 when

concentration k exceeds a critical value k∗ at which flow rate is maximal. An early study of tunneltraffic, however, revealed that a different type of waves exists in real traffic (Edie and Baverez1967). In analyzing the tunnel traffic data, Edie and Baverez (1967) noted that “ small changesin flow may not propagate at a speed equal to the slope of the tangent to a steady-state q-k curveas suggested by the hydrodynamic wave theories of traffic flow. Instead, they are carried along atabout stream speed or only slightly less than stream speed right up to saturation flows, at whichlevel they suddenly reverse directions.” This observation indicates that apart from the family ofwaves that travels against the traffic stream, there’s at least another family of traffic waves thattravels with the traffic stream, even in congested traffic. The findings of Edie and Baverez (1967)prompted Newell (1965) to suggest a fundamental diagram with multiple branches—one branch forfree flow, one for acceleration flow and another for deceleration flow. The extended LWR theorywith this fundamental diagram was able to explain not only the forward wave motion in queued-uptraffic, but also the instability found in tunnel traffic.

In comparison with other deficiencies of the LWR theory, the fourth deficiency, namely its

5-11

inability to model traffic instability, has more serious consequences, for traffic instability is at thevery heart of the traffic congestion phenomenon. The LWR model is always stable in the sensethat traffic disturbances, small or large, are always dampened. In other words, a driver who obeysthe LWR driving law always responds to stimulus properly, i.e., he always manages to changehis speed in the right amount of time and with the right magnitude that he simply absorbs thedisturbance. In fact the reaction time of a LWR driver is zero and the rate of adjustment he makesis infinite. In reality, a driver responds to traffic events with a time delay, and not always precisely.As a result, some disturbances in real traffic may get magnified as they propagate through thetraffic stream, causing traffic break-downs (stop-and-go) that could last for several hours. Suchstop-and-go traffic patterns exhibit, in physical space (i.e., x− t domain), periodic oscillations withamplitude-dependent oscillation time, and in phase space (i.e., q − k or k − u domain), hysteresisloops and wide scatter of data points.

Evidences of traffic instability and the resulting stop-and-go flow pattern are found on highwaysaround the globe. Perhaps the most impressive measurements of transients and stop-start waveformation are gained from European freeways. Due to space restrictions, there are numerousfreeways with two lanes per highway in Europe. These freeways, often equipped with a densemeasurement grid not only for volume and occupancy but also for speed detection, show stable stop-start waves lasting in some cases for more than three hours. Measurement data exists for Germany(Leutzbach 1991), the Netherlands (Verweij 1985), and Italy (Ferrari 1989). First we examine themeasurements from German highways. The German data were collected from the Autobahn A5Karlsruhe-Basel at 617 km by the Institute of Transport Studies at Karlsruhe University (Kuhne1987). Each measurement point is a mean value of a two-minute ensemble actuated every 30 sec.These data were collected during a holiday when no trucks used that stretch of road.

All the data sets have traffic densities over the critical density and show signs of instability (i.e.,stop-start waves with more or less regular shape and of long duration - in some series up to 12traffic breakdowns). These data can characterized in the time domain by their oscillation times T

and magnitudes A. These characteristic features derived from the data shown in Figures 5.9a,band 5.9c,d (in the original text) (Kuhne 1987,Michalopoulos and Pisharody 1980) are listed inthe following table:

Table 1: Oscillation time and magnitudes of stop-and-go traffic from German measurement

oscillation time T 16 min 15 min 7.5 min 5 minamplitude A 70km/h 70 km/h 40 km/h 25 km/hmeasurement figure 2a 2b 3a 3b

These characteristic values show a proportionally between amplitude and oscillation time. Thisstrong dependence is a sign for the non-linear and anharmonic character of stop-start waves. Inthe case of harmonic oscillations, the amplitude is independent of the oscillation time as the linear

5-12


pendulum shows. Obviously, the proportionally holds only for the range between traffic flow at acritical lane speed of about 80 km/h (= speed corresponding to the critical density kc . 25 veh/km)and creeping with jam speed of about 10 km/h. For oscillations covering the whole range betweenfree-flow speed and complete gridlock, saturation effects will reduce the proportionally.

Examples of stop-start waves from other locations, such as the Netherlands (Verweij 1985),Japan (Koshi et al. 1983), Italy (Ferrari 1989) and the U.S.A. also show similar characters asthose observed on German highways. Such oscillations, when viewed from the q − k phase plane,show wider scatter of data points in the congested regime. Embedded in the scatter are sharpdrops (often referred to as the capacity drop) and hysteresis loops (e.g., Treiterer and Myers 1974)of irregular shapes that differ from the equilibrium phase curve q = f∗(k) and cannot be simplyexplained away by stochastic arguments. All but the first deficiencies of the LWR model can beaddressed, to various degrees of success, through the introduction of higher-order approximationsor “dynamic” fundamental diagrams. This leads to two classes of traffic flow models—higher-orderand lower-order continuum models. Higher-order models introduce a dynamic speed-concentrationor flow concentration relation that accounts for driver reaction time and anticipation of trafficconditions ahead, i.e.,

v(x, t + τ) = V∗(k(x + Δx, t)),

q(x, t + τ ) = kV∗(k(x + Δx, t)).

The approximations of these dynamic relations lead to evolution equations for travel speed or flowrate (Payne 1971, Whitham 1974, Zhang 1998). Consequently the traffic model becomes a systemof partial differential equations and its solutions, when shown in the phase plane, deviate fromthe equilibrium fundamental diagram, producing the scatter and forward waves in the congestedregion. Lower-order models, on the other hand, models different traffic motions—acceleration,deceleration and coasting—explicitly on the fundamental diagram. Their fundamental diagramsf∗(k) has multiple branches and connecting curves, each describes a particular kind of motion(Newell 1965, Zhang 2001). For one reason or another, higher-order models are more widely studiedand used than lower-order models. Therefore we will focus our presentation on higher-order modelsin the remaining text. Readers who are interested in lower-order traffic flow models can refer toNewell (1965), Daganzo (1999a,b) and Zhang (2001).

3 Higher-order continuum models

The development of higher-order traffic flow models again originated from the seminal work ofLighthill and Whitham (1955), in which they suggested the following higher-order extension oftheir first-order model:

The LW model: qt + Cqx + Tqtt −Dqxx = 0, (25)

where C is convection speed, T is a reaction time constant and D is the diffusion coefficient.Because of a lack of strong experimental evidence in support of such an extension, higher-order

5-13


approximations of traffic flow were not pursued further till 1971, when Payne (1971) and laterWhitham (1974) derived a so-called ‘momentum equation’ from a car-following argument:

kvt + vvx +c20

kkx =

v∗(k) − v

τ, (26)

where v∗(k) is the equilibrium speed-concentration relation, c0 < 0 is the ‘sound’ speed, and isgiven by c2

0 = μτ , where μ is often referred to as the anticipation coefficient and τ relaxation time

2. Recently, Zhang (1998) proposed a new addition to the existing momentum equations:

vt + vvx + kv′∗(k)2kx =

v∗(k) − v

τ, (27)

which is structurally similar to the momentum equation of Payne (1971) and Whitham (1974).On the left hand side of the momentum equations, the second term is the change of speed

due to convection, and the third term captures drivers’ adjustment to travel speeds owing toanticipation. The term on the right hand side captures drivers’ affinity to equilibrium travel speeds.In the momentum equations, the acceleration of a vehicle, expressed by the material derivative vt +vvx, responds negatively to the increase of concentration downstream, and positively (negatively)to travel speeds that are lower (higher) than the corresponding equilibrium speeds for the sameconcentration. As a result, travel speed v in both momentum equations usually differs from theequilibrium speed v∗(k) under the same traffic condition, but this difference is reduced over timebecause of relaxation effects. The parameter τ decides the strength of relaxation. In literature τ isoften interpreted as driver reaction time, whose value ranges from 1 sec to 1.8 sec.

Through the definition of a concentration dependent sound speed c(k) (= kv′∗(k), or c0, or

−√

μτ ), both momentum equations can be expressed in a general form:

vt + vvx +c2(k)

kkx =

v∗(k) − v

τ. (28)

This evolution equation of travel speed, coupled with the continuity equation

kt + (kv)x = 0, (29)

forms what we call a generalized PW higher-order model. The general PW model comprises asystem of partial differential equations that can be compactly expressed using vector notation

(k

v

)t

+

(v k

c2(k)k v

)(k

v

)x

=

(0

v∗−vτ

). (30)

or

Ut + A(U)Ux = R(U), (31)

where U =

(k

v

), A(U) =

(v k

c2(k)k v

)and R(U) =

(0

v∗−vτ

).

2There are other versions of (26)that differ in values taken by μ. Payne (1971), for example, gives μ = − v′∗2

5-14

This system is strictly hyperbolic because its characteristics, the eigenvalues of the Jacobianmatrix A(U), are real and distinctive

λ1,2 = v ± c(k), λ1 < λ2. (32)

The same is true for the LW model, because, through introduction of two auxiliary variables w = qt

and z = qx the LW model can be transformed into a system of PDEs(

w

z

)t

+

(0 1

−DT 0

)(w

z

)x

= −(

0w + Cz

), (33)

whose characteristics are

λ1,2 = ∓√

D

T, λ1 < λ2.

The two higher-order models differ, however, in that the general PW model is genuinely nonlinearwhile the LW model is linearly degenerate.

Although both the LWR model and the general PW model are hyperbolic, the latter has twocharacteristics, one is always slower than traffic and the other always faster than traffic, owing toc(k) < 0. This is constrasted to the single characteristic of the LWR model, λ∗ = v∗(k) + kv

′∗(k)

that is always slower than traffic. These differences have profound consequences on the behaviorof these models, which we shall discuss in the context of various kinds traffic waves.

3.1 Propagation of traffic sound waves in higher-order models

We first note that traffic sound waves—the propagation of small disturbances in homogeneoustraffic—travel at characteristic speeds in the LWR model. In higher-order models, there are twofamilies of characteristics, thus two characteristic speeds. If a disturbance is still propagated atcharacteristic speeds, then it travels in both directions of traffic with different speeds, reachingdrivers from front and behind. This can be checked through writing the higher-order model inquestion, here the LW model, in a special form⎧⎨

⎩(∂t + C∂x) +

⎛⎝∂t +

√D

T∂x

⎞⎠⎛⎝∂t −

√D

T∂x

⎞⎠⎫⎬⎭ q = 0.

where ∂t + () ∗ ∂x is called a wave operator.From this special form, one can clearly see that the LW model possesses three families of waves:

the first order wave traveling at the convection speed c∗ = C (which is also the characteristic speedof the corresponding first-order model qt + Cqx = 0), the slower second order wave traveling at thefirst characteristic speed c1 = −

√DT and the faster second order wave traveling at the speed of the

second characteristic c2 =√

DT . For certain parameter values of D and T , the second characteristic

speed can be greater than the convection speed, indicating that fast waves reach traffic from behind(the rear view mirror effect). In fact, this is also required for LW traffic to be stable. Otherwise

5-15

the first-order signals would violate the second-order signals, which leads to traffic instability. Thegeneral stability condition for the above higher-order model is

c1 ≤ c∗ ≤ c2, (34)

that is, the first order waves are sandwiched between the two second order waves. This stabilitycondition can be derived from Fourier stability analysis (e.g., Whitham 1974, Kuhne 1984, Zhang1999).

For the general PW model, small perturbations around equilibrium points (k0, v0) would prop-agate in the same way as the waves in the LW model, only with different wave speeds. Instead ofc∗ = C, c1 = −

√DT , c2 =

√DT , here we have c∗ = λ∗(k0), c1 = v0 + c(k0), and c2 = v0 − c(k0).

Because of c(k) < 0, fast waves in the general PW model also reaches vehicles from behind.There are subtle differences among special cases of the general PW model. In the PW model,

where c(k) = c0, the stability condition (34) can be violated and waves grow in magnitude andeventually become shocks in the form of roll waves (Whitham 1974, Kuhne 1984) while in Zhang’smodel where c(k) = kv

′∗(k) the stability condition is always satisfied and waves always damp in

magnitude (Zhang 1999). Thus, like the LWR model, Zhang’s model is also inherently stable.Moreover, the asymptotic behavior of small perturbations of the kind k = k0 + ξ(x, t), v = v0 +w(x, t) near equilibrium point (k0, v0) (v0 = v∗(k0) and ξ,w are small perturbations) in the PW andZhang models also differ. Such perturbations to Zhang’s model can be accurately approximated bya convection equation

ξt + λ∗(k0)ξx = 0, (35)

while those to the PW model can be accurately approximated by a diffusion equation

ξt + λ∗(k0)ξx = D∗ξxx, or (36)

ξt′ = D∗ξx′x′ (37)

if a moving coordinate x′= x − λ∗(k0)t, t

′= t is used, where D∗ is the diffusion coefficient, and

is given by −(λ1(k0, v0) − λ∗(k0))(λ2(k0, v0) − λ∗(k0)), a positive constant (Whitham 1974, delCastillo et al. 1994, and Zhang 1999). Driven by relaxation, any non-equilibrium state (k, v) in thePW model will become closer and closer to its corresponding equilibrium state (k, v∗(k)) with theincrease of time. This latter behavior, together with the properties of Eq. 37 (i.e., its solutions aresmooth), implies that at the end of a queue the PW model does backward smoothing to a sharpdensity/speed profile, thus predicting possibly negative travel speeds (Daganzo 1995a). Zhang’smodel, however, is absent of this problem because its solutions are not diffusive.

3.2 Propagation of shock and expansion waves

When traffic conditions undergo sharp transitions, shocks or expansion waves arise. The LWRtheory possesses both types of waves and whether a particular kind of wave arises is determinedby the entropy condition. A LWR shocks has zero width and travels at a particular speed given by

5-16

the Rankine-Hugoniot condition s = [f∗]/[k], which is derived from the integral conservation law.The general PW model also has similar properties. It has both kinds of waves, actually two foreach kind—one associated with the first characteristic and the other with the second characteristic.The first is referred to as 1-shock or 1-rarefaction waves, and the second 2-shock or 2- rarefactionwaves. Whether a shock or expansion wave arises in a general PW solution is also dependent onentropy conditions, and they are as follows (Zhang 2000a):

1-Shock, (E-H1): λ1(Ur) < s < λ1(Ul), s < λ2(Ur), (38)

2-Shock, (E-H2): λ2(Ur) < s < λ2(Ul), s > λ1(Ul), (39)

1-Rarefaction, (E-R1): λ1(Ul) < λ1(Ur), (40)

2-Rarefaction, (E-R2): λ2(Ul) < λ2(Ur), (41)

And the speeds of shocks are also given by the Rankine-Hugoniot conditions derived fromintegral conservation laws. A difficulty arises, however, from the fact that the momentum equationis not a conservation law, it is simply an evolution equation for travel speeds. As a result, we donot have a unique corresponding integral conservation law for the momentum equation, that is,there are many different integral conservation forms that lead to the momentum equation. Twosuch examples are:

∂

∂t

∫ x2

x1

v(x, t)dx =

(v2

2+ φ(k)

)(x1, t) −

(v2

2+ φ(k)

)(x2, t) (42)

+∫ x2

x1

v∗ − v

τdx, φ

′(k) =

c2(k)k

,

∂

∂t

∫ x2

x1

(kv)(x, t)dx =

((kv)2

k+ ψ(k)

)(x1, t) −

((kv)2

k+ ψ(k)

)(x2, t) (43)

+∫ x2

x1

kv∗ − kv

τdx, ψ

′(k) = c2(k).

We cannot tell, from physical principles, which integral form is “correct”. The selection of aparticular integral form should therefore be guided by field observations, i.e., choosing the one thatproduces closest shock speeds to field measurements. For illustration purposes, we shall take thesimplest, i.e., the first integral form in our subsequent presentations. The same arguments can beapplied directly to other integral forms. The first integral momentum equation leads to followingconservative differential form:

vt +

(v2

2+ φ(k)

)x

=v∗ − v

τ.

together with the conservation of mass

kt + (kv)x = 0,

5-17

we obtain the system of “conservation ” laws with a source term:

Ut + F (U)x = R(U), (44)

where F (U) =(kv, v2

2 + φ(k))t

, and U , R are as defined before.The shock speeds of the general PW model is not affected by the presence of the source term,

and is still given by the Rankine-Hugoniot condition:

s[U ] = [F (U)], (45)

or

s[k] = [kv], s[v] =

[v2

2+ φ(k)

].

These equations imply a certain relation between k and v. This relation leads to the shock curves(1-shock & 2-shock ) in the k − v phase plane (Zhang 2000a).

Expansion (or rarefaction) waves of the general PW model, however, are influenced by therelaxation source term. The strength of this influence is determined by the relaxation time. In ashort time (compared to the relaxation time), one can neglect the effect of the source term andobtain certain relations between k, v in an expansion wave solution. These are the 1-rarefaction and2-rarefaction curves in the k − v phase plane (Zhang 2000a). Over time, however, the cumulativeeffects of the source term, relaxation, build up into rarefaction wave solutions, and one addresses thisproblem by modifying the solutions obtained without the source term. Also because of relaxation,the general PW model either approaches to the LWR model or a viscous LWR model kt +(kv∗)x =D∗kxx, depending on the choice of c(k).

Clearly one cannot ignore the effects of relaxation. Nevertheless, the study of the general PWmodel without relaxation

Ut + F (U)x = 0 (46)

reveals much information about the properties of the general PW model with relaxation. Andcomputations of numerical solutions to the general PW model often rely on the Riemann solutionsof (46). Therefore we provide here the solutions of Riemann problems to the general PW modelwithout relaxation (46) (see Zhang 2000a for details).

The solution to the Riemann problem

Ut + F (U)x = 0, (47)

U (x, t = 0) =

{Ul, x < 0Ur , x > 0

(48)

are of 8 kinds:

1. 1-shock:

H1: vr − vl = −√

2(kl − kr)(φ(kl) − φ(kr))kl + kr

, kl < kr. (49)

5-18

2. 2-shock:

H2: vr − vl = −√

2(kl − kr)(φ(kl) − φ(kr))kl + kr

, kl > kr. (50)

3. 1-rarefaction:R1: vr =

∫kl

c(k)k

dk, kl > kr (51)

4. 2-rarefaction:R2: vr =

∫kl

c(k)k

dk, kl < kr (52)

5. 1-shock + 2-shock:

H1: vm − vl = −√

2(kl − km)(φ(kl) − φ(km))kl + km

, kl < km. (53)

H2: vr − vm = −√

2(km − kr)(φ(km) − φ(kr))km + kr

, km > kr . (54)

6. 1-rarefaction + 2-rarefaction:

R1: vm =∫

kl

c(k)k

dk, kl > km (55)

R2: vr =∫

km

c(k)k

dk, km < kr (56)

7: 1-rarefaction + 2-shock:

H1: vm − vl = −√


, kl < km. (57)

H2: vr − vm = −√

2(km − kr)(φ(km) − φ(kr))km + kr

, km > kr . (58)

8: 1-shock + 2-rarefaction:

H1: vm − vl = −√


, kl < km. (59)

R2: vr =∫

km

c(k)k

dk, km < kr (60)

where Um = (km, vm)t is an intermediate state that provides the transition from a 1-wave to a2-wave.

The transitions are most clearly seen from the phase diagram (Fig. 5). Note that for a stateUl, there are four special curves (H1, R1, H2, R2) emanating from that point, dividing the quarterplane of k − v into four regions: I, II, III, IV. If the downstream Ur in the Riemann problem fallsin region I, the solution would be of Type 6 ( R1+ R2); in region II, Type 7 (R1+ H2), in regionIII, Type 5 (H1+H2), and in region IV, Type 8 (H1 + R2). Because of the entropy conditions, theintermediate states always fall on a 1-wave curve (i.e., R1, H1). When Ur falls on a particular wavecurve, then the transition involves only one kind of wave and no intermediate state is produced.These are the types 1-4 solutions.

5-19

Figure 5: Phase transition diagram in the solution of Riemann problems (adopted from Zhang(2000a))

3.3 Traveling waves, instability and roll waves

Besides traffic sound waves, shock waves and expansion waves, there is another kind of waves inthe general PW theory. This is the so-called traveling waves (Whitham 1974, Zhang 1999). It isa wave that has a smooth profile and travels at a constant speed, and takes the following form:(k, v)(x, t) = (k, v)(χ), χ = x − st (Fig. 7(a)). For the traveling wave solution to arise, a certainstability condition has to be met. When this condition is violated, another kind of waves, the rollwaves, may arise. A roll wave is a series of smooth, monotonic profiles separated by jumps (Fig.6). The existence of roll waves is a direct consequence of traffic instability.

In this section we first obtain the traveling waves, then construct roll waves based on thetraveling wave solution. Recall that a traveling wave is a steady profile with a translation speed s:

(k, v)(x, t) = (k, v)(χ), χ = x − st. (61)

Substitute the traveling wave solution (61) into the general PW model, one obtains

−skχ + (kv)χ = 0, (62)

−svχ + vvχ +c2(k)

kkχ =

v∗ − v

τ, (63)

5-20


which after integration and further substitution of terms become(c2(k) − Q2

k2

)kχ = f∗(k)− Q − ks, (64)

where Q is an integration constant and

Q = k(v − s). (65)

Because the two states at x = ±∞, (k, v) = (k1,2, v1,2) both satisfy (65), one can compute thespeed of the traveling wave

s =k1v1 − k2v2

k1 − k2,

which is the same as the shock speed given by the Rankine-Hugonoit condition.Let h(k) denote the right hand side of (64), then h”(k) = f”(k) < 0. Moreover, h(k) crosses

the k−axis at most twice at ka, kb, ka ≤ kb. These crossing points corresponding to equilibriumconcentration values and in between them h(k) > 0 (Zhang 1999). Since both (k, v) = (k1,2, v1,2)are equilibrium points, k1,2 are also roots of h(k). Moreover, we have k1 < k2, or kχ > 0 from thefact that f∗(k) is strictly concave. Thus, for a smooth profile of k(χ) to exist, we must have

c2(k)− Q2

k2> 0,

which yields the following stability condition

v + c(k) < s < v − c(k). (66)

That is, the traveling wave must travel slower than the fast characteristic and faster than the slowcharacteristic.

When the stability condition is met, the smooth traveling wave profile can be obtained byintegrating

dχ

dk=

c2(k)− Q2

k2

f∗(k) −Q − ks, (67)

which is given by:

χ =∫ k c2(η) − Q2

η2

f∗(η) −Q − ηsdη. (68)

When the stability condition (66) is violated, that is, the traveling wave travels either fasterthan the fast characteristic or slower than the slow characteristic, a smooth profile connecting k1

and k2 is no longer possible and a discontinuity (i.e., shock) must be inserted at the location wherethe wave turns back on itself. This time one still obtains a monotonic profile but with a shockseparates two smooth pieces (Fig. 7(b)), and the speed of the shock is determined by the R-Hshock condition.

There is a special case when the stability condition is not met. This is the case where both thenumerator and denominator of (67) vanishes, which leads to non-smooth and periodic solutions to

5-21

Figure 6: Roll waves in the moving coordinate χ

Figure 7: Traveling waves and shocks in the PW model

5-22


the general PW model, often referred to as roll waves in hydraulic literature. Note that the physicalsolution of these two algebraic equations is Q = −k0c(k0), where Q is the constant flux measuredrelative to the moving coordinate χ and k0 is the critical density that makes both the numeratorand denominator of (67) vanish. It is necessary that (k0, v0) is an equilibrium state, which fixes theshock speed s = v∗(k0) + c(k0). The critical state also fixes the integration constant in the profileequation (68). The roll waves can then be constructed, in the same manner as shown in Dressler(1949), by piecing together the smooth profiles with shocks at appropriate locations, which in turnare determined by the R-H shock conditions. Although the existence of roll wave solutions in thePW model were known to transportation researchers not long after the development of the PWmodel (e.g., Leutzbach 1985), no one to date has obtained specific roll wave solutions to comparewith real world observations of stop-start waves. With the development of Section 3.2, however,this can now be done.

3.4 Summary and Discussions

The PW-like higher-order traffic flow models extend the basic kinematic wave traffic flow modelin two ways: they allow for non-equilibrium phase transitions and introduce instability. Theseare achieved through the addition of a momentum equation that describes speed evolution. Theresulting flow obtained from these higher-order models, however, are not all that different thanthose obtained from the basic kinematic wave model—shocks, for example, exist in both types ofmodels. Moreover, owning to relaxation, similar types of solutions (e.g. shocks or expansion waves)of these two families of models become much similar in long time (typically 10τ).

On the other hand, some significant differences also exist between the two family of models. Forone, higher-order models have two family of characteristics and waves as compared to one family ofthe kinematic wave model. While the first family of waves in the higher-order models behave muchlike those in the kinematic wave model, the second family behaves quite differently—they travelfaster than traffic, and reaches vehicles from behind. This property of the second characteristichas led to doubts about the validity of higher-order traffic models and interesting discussions overthe pros and cons of higher-order approximations of traffic flow in general (e.g., Daganzo 1995a,Papageorgiou 1998, Lebacque 1999, Zhang 2001). Because following vehicles usually cannot forceleading ones to speed up or slow down. fast-than-traffic waves are quite unrealistic when traffic ison a single-lane highway. When traffic is flowing on a multi-lane highway and passing is allowed,such faster-than-traffic waves do arise as a result of 1) lane-changing, or 2) averaging, or both (seeZhang 2000c for details). Another difference between the two types of models is that higher-ordermodels contain traveling wave and roll wave solutions while the KW model does not. Roll wavesolutions are particularly interesting because of their similarity to observed stop-start waves.

Like the kinematic wave model, higher-order models can also be extended to model inhomo-geneous roads. This is particularly straightforward with the PW model— one simply replace thehomogeneous equilibrium speed-density relation v∗(k) with a space-dependent one v∗(k, x). Theextension of higher-order models to model road networks, however, is not as straightforward and is

5-23


a worthwhile research topic for traffic flow researchers.One of the motivations for developing the PW model was to remove shocks from the model

solutions. This, however, is not fulfilled by the higher-order models that we have covered up tonow, although the PW model does admit smooth traveling wave solutions. Higher-order spacederivatives, in the form of kxx or vxx must be introduced to obtain shocks with a structure. Wecall these models diffusive or viscous (higher-order) models. In contrast, PW like models are calledinviscid higher-order models. The next section briefly introduces some of the popular viscous trafficmodels, and a stochastic extension to a particular viscous higher-order model.

4 Diffusive, viscous and stochastic traffic flow models

4.1 Diffusive and viscous traffic flow models

The first diffusive traffic flow model, which was mentioned in the classical book of Whitham (1974)was obtained by considering a flux that is dependent not only on vehicle concentration, but alsoon the concentration gradient:

q = f∗(k) − νkx,

which, after substitution into the conservation equation, leads to

kt + f∗(k)x = νkxx, (69)

where ν is a positive parameter.This diffusive model, when f∗(k) is quadratic, can be manipulated into the following form

ct + ccx = νcxx, c = f′∗(k), (70)

which is the well-known Burger’s equation.Analytical solutions to the Burger’s equation with given initial data can be obtained through

the so-called Cole-Hopf transformation. We refer the reader to Whitham (1974, pp 96-112) for thedetailed solution formulas and only discuss the qualitative properties of various kinds of solutionsto the Burger’s equation. These are:

1. The solution to the initial value problem ct + ccx = νcxx, c(x, t = 0) = F (x) always exists,and is smooth after t > 0.

2. When ν → 0, the solution approaches that of ct + ccx = 0.

3. For initial data F (x) =

{cl x < 0cr x > 0,

cl > cr (A step function), the solution is a traveling

wave c(x − st), with s = c1+c22 . The width of the traveling wave, as measured by the range

where 90% of the change in cl − cr occurs, is proportional to νcl−cr

. As ν → 0, the width ofthe traveling wave becomes nil and the traveling wave becomes a shock of ct + ccx = 0. Thusthe viscous model provides a shock structure to the LWR model.

5-24

4. For initial data F (x) = Aδ(x) (A single hump), the solution is a nonlinear diffusion wave,similar to those of the heat equation ct = νcxx when R = A

2ν � 1.

5. Also, N-waves and periodic waves can be found in the solution with proper initial data.

Clearly, the addition of the second order derivative kxx to the LWR conservation law does twothings to it because of diffusion: 1) it smoothes the shocks of the LWR conservation law, therebyproviding a shock structure, and 2) it guarantees the uniqueness of solution for small ν, therebyproviding a way to pick solutions from the LWR conservation law. This latter property is oftenexploited in numerical computations of shock wave solutions.

The diffusion corrected conservation law of (69), when f∗(k) is not quadratic, can be approx-imated by the Burger’s equation. In fact, the aforementioned properties of the Burger’s equation(except Property #2, which must be modified) are shared by all known diffusion corrected orviscous traffic flow models, including the following popular viscosity-corrected PW model:(

k

v

)t

+

(v k

c2(k)k v

)(k

v

)x

=

(0

v∗−vτ

)+

(0 00 ν

)(k

v

)xx

. (71)

The viscosity-corrected PW model, however, can become unstable in certain ranges of traffic andtherefore has additional properties. These properties, including the collapse of homogeneous trafficunder local and global perturbations and the formation of vehicle-clusters in stop-and-go traffic, arewell documented in (Kerner &Konhauser 1993, 1994; Kerner, Konhauser & Schilke 1995; Kerner andRehborn 1999; Kuhne & Beckschulte 1993). Interested readers are referred to the aforementionedliterature for detailed discussions of these properties.

4.2 Acceleration noise and a stochastic flow model

Same as section 5.3 in the original text.

5 Numerical approximations of continuum models

All of our continuum models are described by partial differential equations, some (i.e, LWR, PWand Zhang) are hyperbolic while others (e.g., the viscous model of (69)) parabolic. Proper ini-tial/boundary conditions must be prescribed to these equations to form a well posed problem. Thesolutions of continuum models involving general initial/boundary conditions are tedious, if notdifficult, to obtain analytically. Numerical procedures are often employed to solve such problems.

Typically, finite difference methods are applied to solve hyperbolic traffic flow models whilefinite difference or finite element methods are used to solve parabolic traffic flow models. Bothmethods start with a discretization of the time-space domain (t ≥ 0,−L < x < L), with thefollowing grid mesh being the most common:

xi = ih, i = 0,±1,±2, · · · , L/h,

tj = jk, j = 0, 1, 2, · · · , T/k.

5-25

where h ≡ Δx and k ≡ Δt, and (xi, tj) are the grid points of this mesh.Let the values of U (x, t) on those grid points denoted by Uj

i (see Fig. 8). Then the time-spacederivatives in a continuum model can be approximated using values at these grid points, and weobtain a set of finite difference equation(s) (FDE). For example, the space derivative Ux can beapproximated in a number of ways:

[Ux]ji =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

Uji −Uj

i−1

h , Forward DifferenceUj

i+1−Uji−1

2h , Center DifferenceUj

i+1−Uji

h , Backward Difference

(72)

Figure 8: Time-space grid

However, the numerical approximation of continuum models is not a simple exercise of replacingcontinuous variables and their derivatives with discrete ones and their differences. This is particu-larly true for hyperbolic traffic models, because in these models discontinuities or shock waves candevelop spontaneously even from smooth initial data. The existence of shocks presents a majorchallenge to the development of numerical approximations that are consistent with and convergentto the model equations when the mesh size is further and further refined. A valid approximationmust meet the following three conditions:

1. it is consistent with the original PDE, i.e., the FDE converges to the PDE when h and κ

approach 0 (consistency),

2. numerical errors introduced by the FDE do not increase over time ( stability), and

3. its solutions converge to the right solutions of the original PDE when h and κ approach 0(convergence).

In this section we present two consistent, stable and convergent finite difference approximationsof the system (31) with initial/boundary data of (73) and a finite element approximation of the

5-26


viscous model (33) with the same initial and boundary data. It should noted that the LWR,PW and Zhang models can all be expressed in the form of (31). Therefore the finite differenceapproximations presented here applies to all three models. We begin our presentation with finitedifference approximations.

5.1 Finite difference methods for solving inviscid models

In this section we present two finite difference schemes to solve the system of (31) with the followinginitial/boundary data:

I.C.: U(x, 0) = U0(x), −L ≤ x ≤ L

B.C.: U(−L, t) = U−(t), U(L, t) = U+(t), t ≥ 0.(73)

where U0 and U± are vector valued functions of space and time, respectively, and may contain acountable number of jumps. This system of equations include the Kinematic wave model of LWR,the higher-order models of Payne-Whitham and of Zhang.

It turns out that a particular form of hyperbolic PDEs, called the conservative form, is speciallysuited for developing finite difference schemes that ensure the aforementioned three conditions. Aconservative form of (31), for example, is given by (44), i.e.,

Ut + F (U)x = R(U).

(44) is called a conservative form because it arises from certain conservative phenomena in convec-tive transport:

∂

∂t

∫L

U (x, t)dx +∫

∂LF (U)dx =

∫L

R(U)dx. (74)

For example, the conservation of vehicles on a finite road segment [x1, x2] leads to a specific caseof (74):

∂

∂t

∫ x2

x1

k(x, t)dx + (kv)(x2, t)− (kv)(x1, t) =∫ x2

x1

r(k, v)dx. (75)

Using the conservative form, we can develop a conservative finite difference approximation of (44).The advantage of a conservative finite difference approximation is that it ensures the correct com-putation of shock speeds. A finite difference approximation of (44) is conservative if it can bewritten as

U j+1i − U j

i

κ+

F (U ji+1, U

ji ) − F (U j

i , U ji−1)

h= R(U j

i+1, Uji , U

ji−1), (76)

where F in (76) is called numerical flux (explained later) 3. When a finite difference scheme isin conservative form, the condition for consistency is particularly simple. It requires that thenumerical flux function satisfies

F (U, U) = F (U).3The argument list of the numerical flux function can involve more than two nodal points, depending on the

required accuracy of the finite difference approximation. (76) uses two nodal values to compute its numerical flux

and is first order accurate.

5-27


Examples of consistent conservative finite difference schemes in traffic flow are the finite differenceschemes of (Michalopoulos, Beskos & Lin (1984)), (Daganzo 1994), and (Lebacque, 1996) in thescalar case (i.e., the LWR model), the finite difference scheme of Leo and Pretty (Leo and Pretty1992) and Zhang (2000b) in the system case (e.g., the PW model).

A consistent, conservative finite difference approximation, if linear, is guaranteed to convergeto the correct solution if it meets a stability condition (LeVeque 1992). This condition, generallyreferred to as the CFL (Courant-Friedrichs-Lewy) condition, says that the cell advance speed h

κ

cannot be greater than the maximum absolute characteristic velocity, i.e.,

max∣∣∣∣κhλi

∣∣∣∣ ≤ 1, i = 1, · · · , n.

Although this theorem has not been proven for most nonlinear systems, computational expe-riences indicate that the CFL condition is sufficient to ensure convergence for a large number ofnonlinear systems. Thus we require the CFL condition in all of our finite difference approximations.

The remaining task in our finite difference approximations is to obtain the numerical fluxfunction that ensures the consistency, stability and convergence properties. Before explaining whata numerical flux function is, we first make clear what U

ji represents in our scheme of things. Suppose

that u(x, t) is a weak solution of the integral conservation law (74). We can write (74) as∫ xi+1/2

xi−1/2

u(x, tj+1)dx =∫ xi+1/2

xi−1/2

u(x, tj)dx +∫ tj+1

tj

∫ xi+1/2

xi−1/2

R(u(x, t))dxdt (77)

−[∫ tj+1

tjF (u(xi+1/2, t)dt −

∫ tj+1

tjF (u(xi−1/2, t)dt

]

where i− 1/2 and i + 1/2 denotes the left and right boundary of cell i (see Fig. 8), respectively.If we interpret Uj

i as the cell average

Uji =

1h

∫ xi+1/2

xi−1/2

u(x, tj)dx (79)

thenF (U j

i+1, Uji ) =

1κ

∫ tj+1

tjF (u(xi+1/2, t)dt (80)

is the average flux passing through the cell boundary xi+1/2 in the time interval (tj , tj+1), and

hR =1κ

∫ tj+1

tj

∫ xi+1/2

xi−1/2

R(u(x, t))dxdt

is the average inflow into cell i from the source during time interval (tj , tj+1). With these definitions,the integral conservation law (74) reduces to (76), and this is why (76) is called a conservativeapproximation.

There are a number of ways to construct a numerical flux function, perhaps the most intuitiveand illustrative is the one obtained using Godunov’s finite difference method. The Godunov methodsolves locally a Riemann problem at each cell boundary for the time interval (tj, tj+1), using the

5-28


cell averages U ji as initial data. It then pieces together these Riemann solutions at time tj+1 and

average them using (79) to obtain new initial data for tj+2, and this process is repeated until T/k

is reached. Recall that the Riemann problem for the homogeneous equation

Ut + F (U)x = 0 (81)

of our continuum traffic models have been solved in Sections 2 and 3. We can then apply theprinciple of superposition to solve the traffic models with source terms:

Ut + F (U)x = R(U). (82)

Using the aforementioned procedure, we obtain a Godunov type of difference equation for (44):

U j+1i − Uj

i

k+

F(U ∗j

i+1/2

)− F

(U∗j

i−1/2

)h

= R. (83)

in which the numerical flux function reads

F (U ji+1, U

ji ) = F

(U

∗ji+1/2

), (84)

and the source flux is computed by

R = R

(Ui+1 + Ui−1

2

).

The variables U ∗ji+1/2, i = 0,±1,±2, · · · ,±L/h are obtained from solving a series of Riemann

problems to the homogeneous equation at cell boundaries. Owing to space limitations, we cannotin this monograph fully describe how this is done for higher-order models (interested readers arereferred to Zhang 2000b). However, the computation of U∗j

i+1/2 is particularly simple when theequilibrium relation v = v∗(k) is assumed. It leads to the following finite difference equation

kj+1i − kj

i

k+

f∗(k∗j

i+1/2

)− f∗

(k∗j

i−1/2

)h

= 0, vj+1i = v∗(k

j+1i ), (85)

where the cell boundary flow f∗(k∗j

i+1/2

)is given by a simple formula (LeVeque 1992, Bui, et. al.

1992):

f∗(ρ∗ji+1/2

)=

⎧⎨⎩

minkji ≤k≤ρj

i+1f∗(k), if k

ji < k

ji+1

maxkj

i≥k≥ρji+1

f∗(k), if kji > kj

i+1

.

When f∗(k) is concave, this formula can be further streamlined through the introduction of a supplyand demand function for each cell (Daganzo 1994, 1995b; Lebacque 1996):

Demand: D(i, j) =

{f∗(k

ji ), if j

i < k∗f(k∗), if j

i ≥ k∗ , (86)

Supply: S(i, j) =

{f∗(k

ji ), if kj

i > k∗f(k∗), if kj

i ≤ k∗ , (87)

5-29

where k∗ is the critical density at which f′∗(k∗) = 0. And we have

f∗(k∗ji+1/2

)= min{Di, Si+1}, (88)

which also applies to boundary cells and bottlenecks.Another difference approximation of (31) uses the idea of Lax-Friedrichs center differencing. It

leads to the following numerical flux:

F (U ji+1, U

ji ) =

F (U ji+1) + F (U j

i )2

− h

κ

U ji+1 + Uj

i

2. (89)

It is easy to check that this flux function meets the consistency requirement and the resulting finitedifference approximation

U j+1i =

Uji+1 + U

ji

2− κ

2h

[F (U j

i+1) − F (U ji−1)

]+ hκR

is also conservative. This center difference scheme remained till recent years a popular choiceof approximation of the kinematic wave model (e.g., Michalopolous 1988, Michalopolous, Beskos& Yamauchi 1984, Michalopolous, Kwon, & Khang 1991, Michalopolous, Lin, & Beskos 1987)and being used lately to approximate higher-order models (Zhang 2000d). Zhang and Wu (1999)investigated the convergence properties of this scheme and found that in comparison with Godunov-type of schemes, the center difference scheme has faster convergence rate with respect to expansionwave solutions and slower convergence rate with respect to shock solutions. The reason is that thisdifference scheme has built-in numerical viscosity which smoothes shocks.

5.2 Finite element methods for solving viscous models

Apart from finite difference methods, the method of finite element is also employed to solve con-tinuum traffic flow equations. In this section we show how the latter is used to solve the viscosity-corrected PW equations.

First we introduce an auxiliary variable w : w = vx, and normalize all the state and time-spacevariables in the following way:

k′ =k

kjamv′ =

v

vfw′ =

w

vfx′ =

x

vfτt′ =

t

τ. (90)

Then the unknown variables in the viscosity-corrected PW model can be expressed by a vector

η =

⎛⎜⎝

k′

v′

w′

⎞⎟⎠ (91)

and the model itself by the following vector-valued quasi-linear partial differential equation:

Aηt + Bηx = C (92)

5-30


with (note that for convenience the ′’s are dropped from the notations)

A =

⎛⎜⎝

1 0 00 1 00 0 0

⎞⎟⎠ B =

⎛⎜⎝

v 0 01k

1Fr 0 1

Re

0 1 0

⎞⎟⎠ C =

⎛⎜⎝

−kw

−vw +v∗ −v

w

⎞⎟⎠ , (93)

Fr ≡ Froude number =kinetic energy (inertia influence)

potential energy(pressure)

=

(12

)kv2

f

c20k

=(

vf

c0

)2

R ≡ Reynolds number =length velocity

kinem. viscosity=

v2fτ

ν0. (94)

The problem also comes with possibly two initial conditions and six boundary conditions. Be-cause of the hyperbolic nature of the viscosity-corrected PW model, however, only certain combi-nations of these initial/boundary data are allowed.

In the finite element method, we replace the continuous functions

η(x, t) =

⎛⎜⎝

k(x, t)u(x, t)w(x, t)

⎞⎟⎠ (95)

by functions defined as a lattice:

η(x0 + iΔx, t0 + jΔt) ≡ ηi,j (96)

and all derivatives by center difference quotients:

ηx → 12Δx

(ηi+1,j+1 − ηi,j+1 + ηi+1,j − ηi,j) (97)

and the function values by the midpoint values:

ηt → 12Δt

(ηi+1,j+1 + ηi,j+1 − ηi+1,j − ηi,j). (98)

η → 14(ηi+1,j+1 + ηi,j+1 + ηi+1,j + ηi,j) (99)

We then do step-wise integration, starting from time step j = 0 and ending at time step j = J , asshown in Fig. (fig 5.16 from original text). To ensure the stability of the numerical procedure, animplicit integration scheme is used to compute the unknown variables ηi,j , ηi+1,j (for notationalsimplicity we’ll drop subscript j in the remaining text of this section). It turns out that theNewtonian iteration procedure is perfectly suited for this purpose. In this procedure the variablesηi, ηi+1 are replaced by an approximation ηi, ˜ηi+1 and the deviations δηi, δηi+1 are computed bylinearizing the original equations. Denoting the deviation vector by

δi ≡⎛⎜⎝

δ ki

δ ui

δ wi

⎞⎟⎠ (100)

5-31


then the basic equations can be written in the form of

Aiδi+1 + Biδi = Ri (101)

with (α =

2Δx

, β =2

Δt, κ =

1Fr

=c20

v2f

)(102)

Ai =

⎛⎜⎝

β + αU + W Kx K

−U ′e(K)− κKx

K2 + ακ 1K β + W + 1 U − αν

0 α −1

⎞⎟⎠ (103)

Bi =

⎛⎜⎝

β − αU + W Kx K

−U ′e(K) − κKx

K2 − ακ 1K β + W + 1 U + αν

0 α −1

⎞⎟⎠ (104)

Ri = −4

⎛⎜⎝

Kt + KxU + K

Ut + UW − Ue(K) + U + κKxK − νWx

Ux −W

⎞⎟⎠ (105)

where abbreviationsK =

14(ki+1 + ki + ki+1,j + ki,j) (106)

Kt =1

2Δt(ki+1 + ki − ki+1,j − ki,j) (107)

are used.Starting with the initial condition as the lowest approximation

ηi = ηi,j=0 ηi−1 = 0 (108)

and using the left boundary condition

ki=0,j , Vi=0,j =⇒ δ0 =

⎛⎜⎝

00

δw0

⎞⎟⎠ (109)

the δi is computed recursively byδi+1 = A−1

i (Ri −Biδi) (110)

as a function of δw0, which in turn is determined by the right boundary condition

vi=I,j =⇒ δI =

⎛⎜⎝

δkI

0δwI

⎞⎟⎠ . (111)

An alternative rearrangement of the deviations δi is possible in order to produce a tridiagonalform which facilitates the fit of the boundary conditions (Kerner and Konhauser, 1993).

5-32


5.3 Applications

5.3.1 Calibration of model parameters with field measurements

All the continuum models discussed in this monograph contain certain static relations and pa-rameters. These relations include, for the LWR model, the fundamental diagram f∗(k), and forhigher-order models, v∗(k). Since f∗(k) = kv∗(k), knowing one would know the other. The pa-rameters include, but not limited to, free flow speed vf , jam wave speed cj , sound speed c0(< 0)(PW model), jam density kjam, critical density kc, capacity qc, and relaxation time τ (PW model).These relations and parameters capture certain fundamental characteristics of the local drivingenvironment and driver population, and have to be calibrated/obtained locally before applicationof the corresponding models. The attainment of the parameters are achieved in two ways: directmeasurement and data fitting. The former, when its cost is acceptable, is always preferred if there’sa a choice of the two.

The interpretations of the parameters, in most cases, are straightforward and intuitive, whichalso suggest ways to measure them directly from field data. Among the various speeds, for example,one can easily measure free flow speed and jam wave speed, but not traffic sound speed c0. For thoseparameters that can be directly measured, to obtain them is a simple matter of data gathering andprocessing and we will not elaborate on them here. Rather, we focus on the calibration of thoseparameters that have confusing interpretations in literature and are difficult to measure directly.These include sound speed c0 and relaxation time τ .

Recall that the definition of traffic sound speed is the speed of sound waves minus the speed oftraffic that carries these sound waves. In the LWR model, the sound wave speeds are f

′∗(k), and

the traffic speed is v∗(k), therefore c0 = f′∗(k)− v∗(k) = kv

′∗(k) is variable. That is, sound speed in

the LWR (and Zhang’s model for that matter) is not a fundamental parameter. The PW model,however, fixes the sound speed c0 as a fundamental parameter and assumes that it is a constant.The latter assumption is questionable because it is unlikely that drivers respond to stimuli withthe same intensity under free-flow and jam traffic conditions. With that being said, we turn ourattention to the possible ways of measuring c0. Recall that λ1,2 = v ± c0 in the PW model, whichsuggests that if we can measure the speed of the slower wave λ1 and traffic speed v, we can thencompute the sound speed c0. This can be done with instrumented vehicles on a single lane-highwaywhere one can measure the acceleration, speed and position of each vehicle in the traffic stream.Clearly, this is a costly way of obtaining sound speed and is rarely done in practice. In reality, c0,together with another parameter τ , is obtained through data fitting.

Before describing the data fitting procedure, we want to reexamine the interpretation of τ , suchthat we have a sense of its range and would know roughly if the value obtained from our data fittingexercise makes sense. Relaxation in traffic flow refers to the process where non-equilibrium trafficapproaches equilibrium traffic overtime4. The pace of this process is controlled by relaxation timeτ . Clearly, τ takes the human reaction time as its lower limit, which is about 1 − 1.8 seconds. Its

4one should not confuse the latter with free-flow traffic, although free-flow traffic is also equilibrium traffic, so is

jam traffic!

5-33

upper limit, in theory, can be infinite. In practice, one never observes congestion that lasts longerthan a day, not to mention infinite. The upper limit of relaxation, one speculates, would be in theorder of a few minutes, the period of a stop-start wave. This speculation tends to be supported byexisting calibration exercises (del Castillo and Benitez 1995, Cremer et al 1993, Kuhne 1991, 1984,Papageorgiou et al 1990) that reported τ values ranging from1.8s to 108s.

The calibration of model parameters through data fitting usually involves the following steps:

1. Data collection: collection of road data in the forms of number of lanes, locations of ramps,and so forth, and traffic data, in the forms of time series data of flux, occupancy and spotspeeds, at various locations,

2. Numerical approximations of the traffic flow model involved,

3. Calibration: obtain the fundamental diagram for each location from measured data, which inturn determines parameters such as free-flow speed, jam density, capacity flux, and jam wavespeed, and obtain other parameters in the model through data fitting.

The process of data fitting involves minimizing some pre-defined performance measures. Onecommon measure is the sum of square errors between model outputs and measured data:

PI = γ1

∫dt (vcal.(d, t) − vmeas.(d, t))2 + γ2

∫dt (kcal.(d, t) − kmeas.(d, t))2 (112)

which is a function of model parameters to be calibrated, e.g.,

PI = PI(c0, τ) (113)

for the PW model.Because of the hyperbolic nature of the continuum traffic flow models, it is crucial to make sure

that the finite difference or finite element approximations are correct and accurate. For this reasonit is advocated that another step be added to the calibration or validation procedure: the step ofchecking the finite difference approximation (Zhang 2001). The best way to check the correctnessof a numerical approximation, apart from theoretical considerations, is to run through benchmarkproblems, such as Riemann problems. In this way one can rid of transient and boundary conditionsthat may hide the inadequacies of the approximation (Zhang 2001).

For the specific examples of calibrating the model parameters, the readers are referred to thefollowing literature:

• del Castillo and Benitez 1995 (A2 Amsterdam-Utrecht, the Netherlands).

• Kuhne and Langbein-Euchner 1993 (A3 Furth-Erlangen near Nuremberg, Germany)

• Papageorgiou et al 1990 (Boulevard Peripherique, Paris, France), and

• Sailer 1996 (Interstate 35W in Minneapolis, Minnesota, U.S.A.).

5-34


5.3.2 Multilane traffic flow dynamics

Numerical examples of a two-lane ring road to be provided by Panos?

5.3.3 Traffic flow on a ring road with a bottleneck5

In this section we use the finite difference approximations of the LWR and PW models developedearlier to simulate traffic on a ring road. The length of the ring road is L = 800l = 22.4 km. Thesimulation time is T = 500τ = 2500 s = 41.7 min. We partition the road [0, L] into N = 100 cellsand the time interval [0, T ] into K = 500 steps. Hence, the length of each cell is Δx = 0.224 kmand the length of each time step is Δt = 5 s. Since λ∗ ≤ vf = 5l/τ , we find the CFL conditionnumber

λ∗Δt

Δx≤ 0.625 < 1.

Moreover, we adopt in this simulation the fundamental diagram used in (Kerner and Konhauser,1994; Herrman and Kerner, 1998) with the following parameters: the relaxation time τ = 5 s; theunit length l = 0.028 km; the free flow speed vf = 5.0l/τ = 0.028 km/s = 100.8 km/h; the jamdensity of a single lane ρj = 180 veh/km/lane; c0 = 2.48445l/τ = 0.014 km/s = 50.0865 km/h;The equilibrium speed-density relationship is therefore

v∗(ρ, a(x)) = 5.0461

⎡⎣(

1 + exp{[ ρ

a(x)ρj− 0.25]/0.06}

)−1

− 3.72 × 10−6

⎤⎦ l/τ,

where a(x) is the number of lanes at location x. The equilibrium functions v∗(ρ, a(x)) and f∗(ρ, a(x))are given in Figure 9.

The first simulation is about the homogeneous LWR model. Here we assume that the ring roadhas single lane everywhere; i.e., a(x) = 1, for x ∈ [0, L] , and use a global perturbation as the initialcondition

ρ(x, 0) = ρh + Δρ0 sin 2πxL , x ∈ [0, L],

v(x, 0) = v∗(ρ(x, 0), 1), x ∈ [0, L],(114)

with ρh = 28 veh/km and Δρ0 = 3 veh/km; and the corresponding initial condition (114) is depictedin Figure 10.

The results are shown as contour plots in Figure 11, from which we observe that initially waveinteractions are strong but gradually the bulge sharpens from behind and expands from front toform a so-called N -wave that travels around the ring with a nearly fixed profile.

In the second simulation we created a bottleneck on the ring road with the following laneconfiguration:

a(x) =

{1, x ∈ [320l, 400l),2, elsewhere .

(115)

5The results in this section are from unpublished material of Jin and Zhang (2000a,b)

5-35

Figure 9: The Kerner-Konhauser model of speed-density and flow-density relations

Figure 10: Initial condition (114) with ρh = 28 veh/km and Δρ0 = 3 veh/km

5-36


Figure 11: Solutions of the homogeneous LWR model with initial condition in Figure 10

5-37


As before, we also use a global perturbation as the initial condition

ρ(x, 0) = a(x)(ρh + Δρ0 sin 2πxL ), x ∈ [0, L],

v(x, 0) = v∗(ρ(x, 0), a(x)), x ∈ [0, L],(116)

with ρh = 28 veh/km/lane and Δρ0 = 3 veh/km/lane (the corresponding initial condition (116) isdepicted in Figure 12).

Figure 12: Initial condition (116) with ρh = 28 veh/km/lane and Δρ0 = 3 veh/km/lane

The results for this simulation are shown in Figure 13, and are more interesting. We observefrom this figure that at first flow increases in the bottleneck to make the bottleneck saturated, thena queue forms upstream of the bottleneck, whose tail propagates upstream as a shock. In the sametime, traffic emerges from the bottleneck accelerates in an expansion wave. After a while, all thecommotion settles and an equilibrium state is reached, where a stationary queue forms upstreamof the bottleneck, whose in/out flow rate equals the capacity of the bottleneck.

The third and fourth simulation runs are for the PW model, where we used the same initial con-ditions for density as in the first and second simulations, respectively, but different initial conditionsfor traffic speeds. These initial conditions (I.C.) are

5-38


Figure 13: Solutions of the inhomogeneous LWR model with initial condition (116)

5-39


I.C. for the third simulation

a(x) = 1, x ∈ [0, L]ρ(x, 0) = ρh + Δρ0 sin 2πx

L , x ∈ [0, L],v(x, 0) = v∗(ρh, 1) + Δv0 sin 2πx

L , x ∈ [0, L].(117)

I.C. for the fourth simulation

a(x) =

{1, x ∈ [320l, 400l)2, elsewhere

ρ(x, 0) = a(x)(ρh + Δρ0 sin 2πxL ), x ∈ [0, L],

v(x, 0) = v∗(ρh, a(x)) + Δv0 sin 2πxL , x ∈ [0, L].

(118)

The parameters are ρh = 28 veh/km, Δρ0 = 3 veh/km and Δv0 = 0.002 km/s.Again we use the same time step and cell size, which yields a CFL number of

λ2Δt

Δx≤ 0.9375 < 1

that ensures numerical stability of our finite difference approximation. The results of these simu-lation runs are shown in Figure 14 and Figure 15 respectively. Note that the PW model solutionfor the homogeneous road is slightly different that the corresponding LWR solution due to non-equilibrium initial speed, but the PW solution soon (about 10τ) looks very much like the LWRsolution. This can be seen more clearly from time-slice plots of vehicle density, speed and flow rateshown in Figure 16. As can been seen from that figure, the solutions are nearly indistinguishableafter t = 140τ . In contrast, the PW solution for the inhomogeneous road, although shows similarpatterns as the the corresponding LWR solution, does not converge to the LWR solution in longtime (see Figure 15 & Figure 17). In long time, both solutions predict the same location of thetail anb head of the queue, but different discharge rate from the queue—traffic leaving the queueat capacity flow rate in the LWR solution, but below capacity flow rate in the PW solution (seeFigure 17). This result highlights not only the differences between the two models, but also theimportance and need for careful experimental validations of these models6.

6The pikes in density that exceed jam density are possibly caused by traffic being unstable near the tail of the

queue.

5-40

Figure 14: Solutions of the PW model with initial condition (117)

5-41


Figure 15: Solutions of the PW model with initial condition (118)

5-42


Figure 16: Comparison of the LWR model and the PW model on a homogeneous ring road: Solidline is used for the LWR model, and dashed line for the PW model

5-43


Figure 17: Comparison of the LWR model and the PW model on an inhomogeneous ring road:Solid line is used for the LWR model, and dashed line for the PW model

5-44


References

Bui, D., Nelson, P. & Narasimhan, S. L. (1992). Computational realizations of the entropy conditionin modeling congested traffic flow, Technical Report, Texas Transportation Research Institute.

Courant, R. & Friedrichs, K. O. ( 1948) , Supersonic flow and shock waves, Interscience Publishers,Inc., New York.

Courant, R. and Hilbert, D. (1962). Methods of mathematical physics, Interscience Publishers, Inc.,New York.

Cremer, M., F. Meiner, and S. Schrieber (1993). On Predictive Schemes in Dynamic ReroutingStrategies. In C. Daganzo (ed). Theory of Transportation and Traffic Flow, pp. 407-462.

Daganzo, C. F. (1994). The cell transmission model: a dynamic representation of highway trafficconsistent with hydrodynamic theory, Transpn. Res. B, Vol. 28B, No.4, 269-287.

Daganzo, C. F. (1995a). Requiem for second-order approximations of traffic flow. Transpn. Res.-B, vol. 29B, No.4, 277-286.

Daganzo, C. F. (1995b). The cell transmission model, Part II: network traffic. Transpn. Res. -B,vol. 29B, No. 2, 79-93.

Daganzo, C. F. (1995c) , ‘A finite difference approximation of the kinematic wave model of trafficflow’, Transportation Research, B 29B(4), 261–276.

Daganzo, C. F. (1997). Fundamentals of Transportation and Traffic Operations. Pergamon, Oxford.

Daganzo, C. F. (1999a). A Behavioral Theory of Multi-Lane Traffic Flow Part I: Long HomogeneousFreeway Sections, ITS Working Paper, UCB-ITS-RR-99-5.

Daganzo, C. F. (1999b). A Behavioral Theory of Multi-Lane Traffic Flow Part II: Merges and theOnset of Congestion”, ITS Working Paper, UCB-ITS-RR-99-6.

del Castillo, J. M. Pintado, P. and F. G. Benitez (1994). The reaction time of drivers and thestability of traffic flow. Transpn. Res.-B. Vol. 28B, No. 1, 35-60.

del Castillo, J. M. and F. G. Benitez (1995). On functional form of the speed-density relationship—I:general theory, II: empirical investigation. Transp. Res. 29B, 373-406.

Dressler, R. F. (1949). Mathematical Solution of the Problem of Roll-Waves in Inclined OpenChannels. Commun. Pure and Appl. Math, 2, 149-194.

Edie, L. C. and E. Bavarez (1967). Generation and propagation of stop-start traffic waves. In L.C. Edie, R. Herman and R. Rothery, eds., Vehicular Traffic Science, Elsevier, 26-37.

5-45


Gazis, D. C., R. Herman, and G. H. Weiss (1962). Density Oscillations Between Lanes of a MultilaneHighway. Oper. Res.,10, 658-667.

Ferrari, P. (1989). The Effect of Driver Behaviour on Motorway Reliability. Transportation Re-search, 23B, 139-150.

Herrmann, M. and B.S. Kerner (1998). Local cluster effect in different traffic flow models, {emPhysica A 255, 163-198.

Jin, W. L. and H. M. Zhang (2000a). The inhomogeneous kinematic wave traffic flow model as aresonant nonlinear system, submitted to Transpn. Sci..

Jin, W. L. and H. M. Zhang (2000b). Numerical studies on the PW model, Working Paper,Institute of Transportation Studies, University of California at Davis.

Kerner, B. S. and Konhauser, P. (1993). Cluster effect in initially homogeneous trafficflow, PhysicalReview E 48(4), 2335–2338.

Kerner,B.S. and P. Konhauser (1994). Structure and parameters of clusters in traffic flow, PhysicalReview E Volume 50, Number 1, 54-83.

Kerner, B. S., Konhauser, P. and M. Shilke (1996). A new approach to problems of traffic flowtheory. in: Proceedings of the 13th Int. Symp. on Transportation and Traffic Theory, J. B.Lesort, editor. 79-102, Pergamon, Oxford, U.K..

Kerner, B. S. and H. Rehborn (1999). Theory of congested traffic flow: self organization withoutbottlenecks. Proceedings of the 14th Int. Symp. on Transportation and Traffic Theory, Cedar,A., editor. 147-171, Pergamon, New York, NY.

Koshi, M., Iwasaki, M. and I. Ohkura (1983). Some findings and an overview on vehicular flowcharacteristics. Proceedings of the 8th Int. Symp. on Transportation and Traffic Theory, Hurdle,V., Hauer, E. and G. Stuart, editors. 403-451, University of Toronto Press, Toronto, Canada.

Kuhne, R. D. (1984). Macroscopic freeway model for dense traffic— stop-start waves and incidentdetection, Ninth International Symposium on Transportation and Traffic Theory, VNU SciencePress, 20-42.

Kuhne, R. D. (1987). Freeway Speed Distribution and Acceleration Noise in N. H. Gartner, N. H.M. Wilson (eds.). Proceedings of the Tenth International Symposium on Transportation andTraffic Theory, Elsevier, 119-137.

Kuhne, R. D. (1989). Freeway control and incident detection using a stochastic continuum theory oftraffic flow. In Proc. 1st int. conf. on applied advanced technology in transportation engineering,pages 287–292, San Diego, CA.

5-46


Kuhne, R. D. and Beckschulte, R. (1993). Non-linearity stochastics of unstable traffic flow, in C. F.Daganzo, ed., ‘Transportation and Traffic Theory’, Elsevier Science Publishers., pp. 367–386.

Kuhne, R. D. and K. Langbein-Euchner (1995). Parameter Validation. Report as an Orderof Daimler-Benz Research, Stuttgart. Available by Steierwald Schnharting und Partner,Stuttgart.

Lax, P. D. (1972). Hyperbolic systems of conservations laws and the mathematical theory of shockwaves, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania.

Lebacque, J. P. (1996). The Godunov scheme and what it means for first order traffic flow models.Proceedings of the 13th Int. Symp. on Transportation and Traffic Theory, J-P Lesort, editor.647-677, Pergamon, New York, NY.

Lebacque, J. P. (1999). Macroscopic traffic flow models: a question of order. Proceedings of the14th Int. Symp. on Transportation and Traffic Theory, Cedar, A., editor. 147-171, Pergamon,New York, NY.

Leo, C. J., and R. L. Pretty (1992). Numerical simulations of macroscopic continuum traffic models.Transpn. Res.-B. 26B, No.3, 207-220.

Leutzbach, W. (1985). Introduction to the Theory of Traffic Flow. Springer Pub., 184-193.

Leutzbach, W. (1991) Measurements of Mean Speed Time Series Autobahn A5 near Karlsruhe,Germany. Institute of Transport Studies, University of Karlsruhe, Germany.

LeVeque, R. (1992) Numerical methods for conservation laws, Birkhauser Verlag.

Lighthill, M. J. and Whitham, G. B. (1955). On kinematic waves: II. a theory of traffic flow onlong crowded roads. In Proc. Royal Society, volume 229(1178) of A, 317–345.

Michalopoulos, P. G., Beskos, D. E. & Lin, J. K. (1984) , ‘Analysis of interrupted traffic flow byfinite difference methods’, Transportation Research, B 18B, 377–396.

Michalopoulos, P. G., D. E. Beskos, and Y. Yamauchi, (1984). Multilane Traffic Flow Dynamics:Some Macroscopic Considerations. Transportation Research, 18B, No. 4/5, pp. 377-395.

Michalopoulos, P. G., Jaw Kuan Lin, and D. E. Beskos, (1987). Integrated Modelling and NumericalTreatment of Freeway Flow. Appl. Mathem. Model, Vol. 11, No. 401, 447-458.

Michalopoulos, P. G., E. Kwon, and E. G. Khang, (1991). Enhancements and Field Testing of aDynamic Simulation Program. Transportation Research Record, 1320, pp. 203-215.

Newell, G. F. (1961). Nonlinear Effects in the Dynamics of Car Following, Operations Research,Vol 9, pp.209-229.

5-47


Newell, G. F. (1965). Instability in dense highway traffic, a review. In Almond, P., editor, Proceed-ings of the Second International Symposium on the Theory of Traffic Flow, pages 73–85.

Newell, G. F. (1993). A simplified theory of kinematic waves in highway traffic, I general theory, IIqueuing at freeway bottlenecks, III multi-destination flows. Transpn. Res. B, Vol. 27, 281-313.

Papageorgiou, M., Blosseville, J., and Hadj-Salem, H.(1990) Modeling and Real-Time Controlof Traffic Flow on The Southern Part of Boulevard Peripherique in Paris: Part I: Modeling,Transportation Research, A, 24 (5) 345-359.

Papageorgiou, M. (1998). Some remarks on macroscopic flow modeling. Transportation Research,A, 32 (5) 323-329.

Payne, H. J. (1971). Models of freeway traffic and control. In Bekey, G. A., editor, MathematicalModels of Public Systems, volume 1 of Simulation Councils Proc. Ser., 51–60.

Payne, H. J. (1979). FREFLO: A Macroscopic Simulation Model of Freeway Traffic. TransportationResearch Record 722, pp. 68-77.

Richards, P. I. (1956). Shock waves on the highway. Operations Research, 4:42–51.

Sailer, H. (1996). Fuid Dynamical Modelling of Traffic Flow on Highways–Physical Basis andNumerical Examples; Dissertation. (submitted for a Diploma) at the Institute for TheoreticalPhysics, University Innsbruck.

Treiterer, J. and J. A. Myers (1974). The hysteresis phenomena in traffic flow. Proceedings of thesixth Int. Symp. on Transportation and Traffic Theory, D. J. Buckley, editor. 13-38.

Verweij, H. D. (1985). Filewaarschuwing en Verkeer-safwikkeling. Ministry of Traffic and Water-Ways, Delft, The Netherlands.

Whitham, G. B. (1974). Linear and nonlinear waves. John Wiley & Sons, New York.

Zhang, H. M. (1998). A theory of nonequilibrium traffic flow. Transportation Research, B,32(7):485–498.

Zhang, H. M. (1999). Analyses of the stability and wave properties of a new continuum traffictheory. Transportation Research, B, 33 (6) 399-415.

Zhang, H. M. and T. Wu (1999) Numerical simulation and analysis of traffic flow. TransportationResearch Record 1678, 251-260.

Zhang, H. M. (2000a). Structural properties of solutions arising from a nonequilibrium traffic flowtheory. Transportation Research, B. 34, 583-603 .

Zhang, H. M. (2000b). A finite difference approximation of a non-equilibrium traffic flow model.Transportation Research, B. (in press).

5-48


Zhang, H. M. (2000c). Anisotropic property revisited—when is it violated in traffic flow? Workingpaper, Institute of Transportation Studies, University of California at Davis (accepted forpublication in Trans. Res., -B).

Zhang, H. M. (2000d). Phase transitions in a non-equilibrium traffic theory, ” ASCE Journal ofTransportation Engineering. Vol. 126, No. 1, 1-12.

Zhang, H. M. and T. Kim (2000). Effects of relaxation and anticipation on Riemann solutions ofthe PW model-a numerical investigation. Transportation Research Record (in press).

Zhang, H. M. (2001). New perspectives on continuum traffic flow models. Special issue on trafficflow theory, H. M. Zhang, ed., Journal of Networks and Spatial Economics, Vol. 1, Issue 1.2001.

5-49


Associate Professor, Department of Civil Engineering, University of Texas at Arlington, Box 19308,9

Arlington, TX 76019-0308.

MACROSCOPIC FLOW MODELS

BY JAMES C. WILLIAMS9


Note to reader: The symbols used in Chapter 6 are the same as those used in the original sources. Therefore, the reader is cautioned thatthe same symbol may be used for different quantities in different sections of this chapter. The symbol definitions below include the sectionsin which the symbols are used if the particular symbol definition changes within the chapter or is a definition particular to this chapter.In each case, the symbols are defined as they are introduced within the text of the chapter. Symbol units are given only where they helpdefine the quantity; in most cases, the units may be in either English or metric units as necessary to be consistent with other units in arelation.

A = area of town (Section 6.2.1)c = capacity (vehicles per unit time per unit width of road) (Section 6.2.1)D = delay per intersection (Section 6.2.2)f = fraction of area devoted to major roads (Section 6.1.1)f = fraction of area devoted to roads (Section 6.2.1)f = number of signalized intersections per mile (Section 6.2.2)f = fraction of moving vehicles in a designated network (Section 6.3)r

f = fraction of stopped vehicles in a designated network (Section 6.3)s

f = minimum fraction of vehicles stopped in a network (Section 6.4)s,min

I = total distance traveled per unit area, or traffic intensity (pcu/hour/km) (Sections 6.1.1 and 6.2.3)J = fraction of roadways used for traffic movement (Section 6.2.1)K = average network concentration (ratio of the number of vehicles in a network and the network length, Section 6.4)K = jam network concentration (Section 6.4)j

N = number of vehicles per unit time that can enter the CBD (Section 6.2.1)n = quality of traffic indicator (two-fluid model parameter, Section 6.3)Q = capacity (pcu/hr) (Section 6.2.2)Q = average network flow, weighted average over all links in a designated network (Section 6.4)q = average flow (pcu/hr)R = road density, i.e., length or area of roads per unit area (Section 6.2.3)r = distance from CBDT = average travel time per unit distance, averaged over all vehicles in a designated network (Section 6.3)T = average minimum trip time per unit distance (two-fluid model parameter, Section 6.3)m

T = average moving (running) time per unit distance, averaged over all vehicles in a designated network (Section 6.3)r

T = average stopped time per unit distance, averaged over all vehicles in a designated network (Section 6.3)s

V = average network speed, averaged over all vehicles in a designated network (Section 6.4)V = network free flow speed (Section 6.4)f

V = average maximum running speed (Section 6.2.3)m

V = average speed of moving (running) vehicles, averaged over all in a designated network (Section 6.3)r

v = average speedv = weighted space mean speed (Section 6.2.3)v = average running speed, i.e., average speed while moving (Section 6.2.2)r

w = average street width� = Zahavi’s network parameter (Section 6.2.3)� = g/c time, i.e. ration of effective green to cycle length

� � �

6.MACROSCOPIC FLOW MODELS

Mobility within an urban area is a major component of that area's level provides this measurement in terms of the three basicquality of life and an important issue facing many cities as they variables of traffic flow: speed, flow (or volume), andgrow and their transportation facilities become congested. There concentration. These three variables, appropriately defined, canis no shortage of techniques to improve traffic flow, ranging also be used to describe traffic at the network level. Thisfrom traffic signal timing optimization (with elaborate, description must be one that can overcome the intractabilities ofcomputer-based routines as well as simpler, manual, heuristic existing flow theories when network component interactions aremethods) to minor physical changes, such as adding a lane by the taken into account.elimination of parking. However, the difficulty lies in evaluatingthe effectiveness of these techniques. A number of methods The work in this chapter views traffic in a network from acurrently in use, reflecting progress in traffic flow theory and macroscopic point of view. Microscopic analyses run into twopractice of the last thirty years, can effectively evaluate changes major difficulties when applied to a street network:in the performance of an intersection or an arterial. But adilemma is created when these individual components, 1) Each street block (link) and intersection are modeledconnected to form the traffic network, are dealt with collectively. individually. A proper accounting of the interactions

The need, then, is for a consistent, reliable means to evaluate case of closely spaced traffic signals) quickly leads totraffic performance in a network under various traffic and intractable problems.geometric configurations. The development of suchperformance models extends traffic flow theory into the network 2) Since the analysis is performed for each networklevel and provides traffic engineers with a means to evaluate component, it is difficult to summarize the results in asystem-wide control strategies in urban areas. In addition, the meaningful fashion so that the overall network performancequality of service provided to the motorists could be monitored can be evaluated.to evaluate a city's ability to manage growth. For instance,network performance models could also be used by a state Simulation can be used to resolve the first difficulty, but theagency to compare traffic conditions between cities in order to second remains; traffic simulation is discussed in Chapter 10.more equitably allocate funds for transportation systemimprovements. The Highway Capacity Manual (Transportation Research Board

The performance of a traffic system is the response of that service, yet does not address the problem at the network level.system to given travel demand levels. The traffic system consists While some material is devoted to assessing the level of serviceof the network topology (street width and configuration) and the on arterials, it is largely a summation of effects at individualtraffic control system (e.g., traffic signals, designation of one- intersections. Several travel time models, beginning with theand two-way streets, and lane configuration). The number of travel time contour map, are briefly reviewed in the next section,trips between origin and destination points, along with the followed by a description of general network models in Sectiondesired arrival and/or departure times comprise the travel 6.2. The two-fluid model of town traffic, also a general networkdemand levels. The system response, i.e., the resulting flow model, is discussed separately in Section 6.3 due to the extent ofpattern, can be measured in terms of the level of service the model's development through analytical, field, and simulationprovided to the motorists. Traffic flow theory at the intersection studies. Extensions of the two-fluid model into general networkand arterial models are examined in Section 6.4, and the chapter references

between adjacent network components (particularly in the

1994) is the basic reference used to evaluate the quality of traffic

are in the final section.

6.1 Travel Time Models

Travel time contour maps provide an overview of how well a dispatched away from a specified location in the network, andstreet network is operating at a specific time. Vehicles can be each vehicle's time and position noted at desired intervals.

I � A exp � r/a

f � B exp � r/b ,

��

� � �

(6.1)

(6.2)

Contours of equal travel time can be established, providing case, general model forms providing the best fit to the data wereinformation on the average travel times and mean speeds overthe network. However, the information is limited in that thetravel times are related to a single point, and the study wouldlikely have to be repeated for other locations. Also, substantialresources are required to establish statistical significance. Mostimportantly, though, is that it is difficult to capture networkperformance with only one variable (travel time or speed in thiscase), as the network can be offering quite different levels ofservice at the same speed.

This type of model has be generalized by several authors toestimate average network travel times (per unit distance) orspeeds as a function of the distance from the central businessdistrict (CBD) of a city, unlike travel time contour maps whichconsider only travel times away from a specific point.

6.1.1 General Traffic Characteristics as a Function of the Distance from the CBD

Vaughan, Ioannou, and Phylactou (1972) hypothesized severalgeneral models using data from four cities in England. In each

selected. Traffic intensity (I, defined as the total distancetraveled per unit area, with units of pcu/hour/km) tends todecrease with increasing distance from the CBD,

where r is the distance from the CBD, and A and a areparameters. Each of the four cities had unique values of A anda, while A was also found to vary between peak and off-peakperiods. The data from the four cities is shown in Figure 6.1.A similar relation was found between the fraction of the areawhich is major road (f) and the distance from the CBD,

where b and B are parameters for each town. Traffic intensityand fraction of area which is major road were found to belinearly related, as was average speed and distance from theCBD. Since only traffic on major streets is considered, these

Figure 6.1Total Vehicle Distance Traveled Per Unit Area on Major Roads as a

Function of the Distance from the Town Center (Vaughan et al. 1972).

v � a r b

v � c � ar b ,

v � a � br .

v � a � be �cr ,

v �

1�b 2 r 2

a�cb 2 r 2

��

� � �

(6.3)

(6.4)

(6.5)

(6.6)

(6.7)

results are somewhat arbitrary, depending on the streets selected had been fitted to data from a single city (Angel and Hymanas major. 1970). The negative exponential asymptotically approaches

6.1.2 Average Speed as a Function ofDistance from the CBD

Branston (1974) investigated five functions relating averagespeed (v) to the distance from the CBD (r) using data collectedby the Road Research Laboratory (RRL) in 1963 for six cities inEngland. The data was fitted to each function using least-squares regression for each city separately and for the aggregateddata from all six cities combined. City centers were defined asthe point where the radial streets intersected, and the journeyspeed in the CBD was that found within 0.3 km of the selectedcenter. Average speed for each route section was found bydividing the section length by the actual travel time(miles/minute). The five selected functions are described below,where a, b, and c are constants estimated for the data. A powercurve,

was drawn from Wardrop's work (1969), but predicts a zerospeed in the city center (at r = 0). Accordingly, Branston alsofitted a more general form,

where c represents the speed at the city center.

Earlier work by Beimborn (1970) suggested a strictly linearform, up to some maximum speed at the city edge, which wasdefined as the point where the average speed reached itsmaximum (i.e., stopped increasing with increasing distance fromthe center). None of the cities in Branston's data set had a clearmaximum limit to average speed, so a strict linear function alonewas tested:

A negative exponential function,

some maximum average speed.

The fifth function, suggested by Lyman and Everall (1971),

also suggested a finite maximum average speed at the cityoutskirts. It had originally be applied to data for radial and ringroads separately, but was used for all roads here.

Two of the functions were quickly discarded: The linear model(Equation 6.5) overestimated the average speed in the CBDs by3 to 4 km/h, reflecting an inability to predict the rapid rise inaverage speed with increasing distance from the city center. Themodified power curve (Equation 6.4) estimated negative speedsin the city centers for two of the cities, and a zero speed for theaggregated data. While obtaining the second smallest sum ofsquares (negative exponential, Equation 6.6, had the smallest),the original aim of using this model (to avoid the estimation ofa zero journey speed in the city center) was not achieved.

The fitted curves for the remaining three functions (negativeexponential, Equation 6.6; power curve, Equation 6.3; andLyman and Everall, Equation 6.7) are shown for the data fromNottingham in Figure 6.2. All three functions realisticallypredict a leveling off of average speed at the city outskirts, butonly the Lyman-Everall function indicates a leveling off in theCBD. However, the power curve showed an overall better fitthan the Lyman-Everall model, and was preferred.

While the negative exponential function showed a somewhatbetter fit than the power curve, it was also rejected because of itsgreater complexity in estimation (a feature shared with theLyman-Everall function). Truncating the power function atmeasured downtown speeds was suggested to overcome itsdrawback of estimating zero speeds in the city center. Thecomplete data set for Nottingham is shown in Figure 6.3,showing the fitted power function and the truncation at r = 0.3km.

��

� � �

Figure 6.2Grouped Data for Nottingham Showing Fitted a) Power Curve,

b) Negative Exponential Curve, and c) Lyman-Everall Curve (Branston 1974, Portions of Figures 1A, 1B, and 1C).

Figure 6.3Complete Data Plot for Nottingham; Power Curve

Fitted to the Grouped Data (Branston 1974, Figure 3).

��

� � �

If the data is broken down by individual radial routes, as shown Hutchinson (1974) used RRL data collected in 1967 from eightin Figure 6.4, the relation between speed and distance from the cities in England to reexamine Equations 6.3 and 6.6 (powercity center is stronger than when the aggregated data is curve and negative exponential) with an eye towards simplifyingexamined. them.

Figure 6.4Data from Individual Radial Routes in Nottingham,

Best Fit Curve for Each Route is Shown (Branston 1974, Figure 4).

v � k r 1/3v � 60 � ae �r/R .

N � �f c A,

q � 2440 � 0.220 v 3 ,

c � 58.2 � 0.00524 v 3 .

��

� � �

(6.8)(6.9)

(6.10)

(6.11)

(6.12)

The exponents of the power functions fitted by Branston (1974) city. Assuming that any speed between 50 and 75 km/h wouldfell in the range 0.27 to 0.36, suggesting the following make little difference, Hutchinson selected 60 km/h, andsimplification

When fitted to Branston's data, there was an average of 18 30 percent (on the average) over the general form used bypercent increase in the sum of squares. The other parameter, k, Branston. R was found to be strongly correlated with the citywas found to be significantly correlated with the city population, population, as well as showing different averages with peak andwith different values for peak and off-peak conditions. Theparameter k was found to increase with increasing population, population nor the peak vs. off-peak conditions. The differenceand was 9 percent smaller in the peak than in the off-peak.

In considering the negative exponential model (Equation 6.6),Hutchinson reasoned that average speed becomes lesscharacteristic of a city with increasing r, and, as such, it would used RRL data collected in 1967 from eight cities in England tobe reasonable to select a single maximum limit for v for every

Hutchinson found that this model raised the sum of squares by

off-peak conditions, while a was correlated with neither the city

in the Rs between peak and off-peak conditions (30 percenthigher during peaks) implies that low speeds spread out overmore of the network during the peak, but that conditions in thecity center are not significantly different. Hutchinson (1974)

reexamine Equations 6.3 and 6.6 (power curve and negativeexponential) with an eye towards simplifying them.

6.2 General Network Models

A number of models incorporating performance measures otherthan speed have been proposed. Early work by Wardrop andSmeed (Wardrop 1952; Smeed 1968) dealt largely with thedevelopment of macroscopic models for arterials, which werelater extended to general network models.

6.2.1 Network Capacity

Smeed (1966) considered the number of vehicles which can"usefully" enter the central area of a city, and defined N as thenumber of vehicles per unit time that can enter the city center.In general, N depends on the general design of the road network,width of roads, type of intersection control, distribution ofdestinations, and vehicle mix. The principle variables for townswith similar networks, shapes, types of control, and vehicles are:A, the area of the town; f, the fraction of area devoted to roads;and c, the capacity, expressed in vehicles per unit time per unitwidth of road (assumed to be the same for all roads). These arerelated as follows:

where � is a constant. General relationships between f and(N/c�A) for three general network types (Smeed 1965) areshown in Figure 6.5. Smeed estimated a value of c (capacity perunit width of road) by using one of Wardrop's speed-flowequations for central London (Smeed and Wardrop 1964),

where v is the speed in kilometers/hour, and q the average flowin pcus/hour, and divided by the average road width, 12.6meters,

A different speed-flow relation which provided a better fit forspeeds below 16 km/h resulted in c = 68 -0.13 v (Smeed 1963).2

Equation 6.12 is shown in Figure 6.6 for radial-arc, radial, andring type networks for speeds of 16 and 32 km/h. Data fromseveral cities, also plotted in Figure 6.6, suggests that �c=30,

��

� � �

Note: (e=excluding area of ring road, I=including area of ring road)

Figure 6.5Theoretical Capacity of Urban Street Systems (Smeed 1966, Figure 2).

Figure 6.6Vehicles Entering the CBDs of Towns Compared with the Corresponding

Theoretical Capacities of the Road Systems (Smeed 1966, Figure 4).

N � 33� 0.003 v 3 f A ,

N � 33� 0.003 v 3 J f A

��

� � �

(6.13)

(6.14)

and using the peak period speed of 16 km/h in central London,Equation 6.10 becomes

where v is in miles/hour and A in square feet. It should be notedthat f represents the fraction of total area usefully devoted toroads. An alternate formulation (Smeed 1968) is

where f is the fraction of area actually devoted to roads, while Jis the fraction of roadways used for traffic movement. J wasfound to range between 0.22 and 0.46 in several cities inEngland. The large fraction of unused roadway is mostly due tothe uneven distribution of traffic on all streets. The number ofvehicles which can circulate in a town depends strongly on theiraverage speed, and is directly proportional to the area of usableroadway. For a given area devoted to roads, the larger thecentral city, the smaller the number of vehicles which cancirculate in the network, suggesting that a widely dispersed townis not necessarily the most economical design.

6.2.2 Speed and Flow Relations

Thomson (1967b) used data from central London to develop alinear speed-flow model. The data had been collected onceevery two years over a 14-year period by the RRL and theGreater London Council. The data consisted of a network-wideaverage speed and flow each year it was collected. The averagespeed was found by vehicles circulating through central Londonon predetermined routes. Average flows were found by firstconverting measured link flows into equivalent passengercarunits, then averaging the link flows weighted by theirrespective link lengths. Two data points (each consisting of anaverage speed and flow) were found for each of the eight yearsthe data was collected: peak and off-peak.

Plotting the two points for each year, Figure 6.7, resulted in aseries of negatively sloped trends. Also, the speed-flow capacity(defined as the flow that can be moved at a given speed)gradually increased over the years, likely due to geometric andtraffic control improvements and "more efficient vehicles." Thisindicated that the speed-flow curve had been gradually changing,indicating that each year's speed and flow fell on different curves.Two data points were inadequate to determine the shape of thecurve, so all sixteen data points were used by accounting for the

Figure 6.7Speeds and Flows in Central London, 1952-1966, Peak and Off-Peak (Thomson 1967b, Figure 11)

v � 30.2 � 0.0086 q

��

� � �

(6.15)

changing capacity of the network, and scaling each year's flow The equation implies a free-flow speed of about 48.3 km/hmeasurement to a selected base year. Using linear regression, however, there were no flows less than 2200 pcu/hour in thethe following equation was found: historical data.

where v is the average speed in kilometers/hour and q is theaverage flow in pcu/hour. This relation is plotted in Figure 6.8.

Thomson used data collected on several subsequent Sundays(Thomson 1967a) to get low flow data points. These arereflected in the trend shown in Figure 6.9.Also shown is a curvedeveloped by Smeed and Wardrop using data from a single yearonly.

Note: Scaled to 1964 equivalent flows.Figure 6.8

Speeds and Scaled Flows, 1952-1966 (Thomson 1967b, Figure 2).

Figure 6.9Estimated Speed-Flow Relations in Central London

(Main Road Network) (Thomson 1967b, Figure 4).

v � 24.3 � 0.0075 q ,

v � 34.0 � 0.0092 q .

1v�

1vr� f d ,

1v�

1a (1�q/Q)

�

f b1�q/�s

vr � 31� 0.70q�4303w

��

� � ��

(6.16)

(6.17)

(6.18)

(6.19)

(6.20)

The selected area of central London could be broken into innerand outer zones, distinguished principally by traffic signaldensities, respectively 7.5 and 3.6 traffic signals per route-mile.Speed and flow conditions were found to be significantlydifferent between the zones, as shown in Figure 6.10, and for theinner zone,

and for the outer zone,

Wardrop (1968) directly incorporated average street width andaverage signal spacing into a relation between average speed andflow, where the average speed includes the stopped time. Inorder to obtain average speeds, the delay at signalizedintersections must be considered along with the running speedbetween the controlled intersections, where running speed isdefined as the average speed while moving. Since speed is theinverse of travel time, this relation can be expressed as:

where v is the average speed in mi/h, v , the running speed inr

mi/h, d the delay per intersection in hours, and f the number ofsignalized intersections per mile. Assuming v = a(1-q/Q) andr

d = b/(1-q/�s), where q is the flow in pcu/hr, Q is the capacityin pcu/hr, � is the g/c time, and s is the saturation flow in pcu/hr,and combining into Equation 6.18,

Using an expression for running speed found for central London(Smeed and Wardrop 1964; RRL 1965),

Figure 6.10Speed-Flow Relations in Inner and Outer Zones of Central Area

(Thomson 1967a, Figure 5).

1v�

128� 0.0058 q

�

0.0057

1� q2610

.

1v�

128� 0.0058 q

�

1197� 0.0775 q

vr � 31 � 4303w

�

a qw

� 31� 140w

�

a qw

.

vr � 31 � 140w

� 0.0244 qw

.

f d �

f b1� q /k�w

f d �

f b1� q /147 �w

f d �

f1000 � 6.8 q /�w

.

1v�

1

31� 140w

� 0.0244 qw

�

f

1000 � 6.8 q�w

.

��

� � ��

(6.21)

(6.22)

(6.23)

(6.24)

(6.25)

(6.26)

where w is the average roadway width in feet, and an averagestreet width of 42 feet (in central London), Equation 6.20becomes v = 28 - 0.0056 q, or 24 mi/h, whichever is less. Ther

coefficient of q was modified to 0.0058 to better fit the observedrunning speed.

Using observed values of 0.038 hours/mile stopped time, 2180pcu/hr flow, and 2610 pcu/hr capacity, the numerator of thesecond term of Equation 6.19 (fb) was found to be 0.0057.Substituting the observed values into Equation 6.19,

Simplifying,

Revising the capacity to 2770 pcu/hour (to reflect 1966 data),thus changing the coefficient of q in the second term of Equation6.21 to 0.071, this equation provided a better fit than Thomson'slinear relation (Thomson 1967b) and recognizes the knowninformation on the ultimate capacity of the intersections.

Generalizing this equation for urban areas other than London,and knowing that the average street width in central London was12.6 meters, the running speed can be written

Since a/w = 0.0058 when w = 42 by Equation 6.21, a=0.0244,then

For the delay term, five controlled intersections per mile and ag/c of 0.45 were found for central London. Additionally, theintersection capacity was assumed to be proportional to theaverage stop line width, given that it is more than 5 meters wide(RRL 1965), which was assumed to be proportional to theroadway width. The general form for the delay equation (secondterm of Equation 6.21) is

where k is a constant. For central London, w = 42, � = 0.45,and k�w = Q = 2770, thus k = 147, yielding

Given that f = 5 signals/mile and fb = 0.00507 for centralLondon, b = 0.00101, yielding

Combining, then, for the general equation for average speed:

The sensitivity of Equation 6.26 to flow, average street width,number of signalized intersections per mile, and the fraction ofgreen time are shown in Figures 6.11, 6.12, and 6.13. Bycalibrating this relation on geometric and traffic control featuresin the network, Wardrop extended the usefulness of earlier speedflow relations. While fitting nicely for central London, theapplicability of this relation to other cities in its generalizedformat (Equation 6.26) is not shown, due to a lack of availabledata.

��

� � ��

Figure 6.11Effect of Roadway Width on Relation Between Average (Journey)

Speed and Flow in Typical Case (Wardrop 1968, Figure 5).

Figure 6.12Effect of Number of Intersections Per Mile on Relation Between

Average (Journey) Speed and Flow in Typical Case (Wardrop 1968, Figure 6).

I � �(v/R)m ,

I � � R/v ,

��

� � ��

(6.27)

(6.28)

Figure 6.13Effect of Capacity of Intersections on Relation Between

Average (Journey) Speed and Flow in Typical Case (Wardrop 1968, Figure 7).

Godfrey (1969) examined the relations between the average �-relationship, below, and the two-fluid theory of town traffic.speed and the concentration (defined as the number of vehicles The two-fluid theory has been developed and applied to a greaterin the network), shown in Figure 6.14, and between average extent than the other models discussed in this section, and isspeed and the vehicle miles traveled in the network in one hour, described in Section 6.3.shown in Figure 6.15. Floating vehicles on circuits within thenetwork were used to estimate average speed and aerialphotographs were used to estimate concentration.

There is a certain concentration that results in the maximum flow(or the maximum number of miles traveled, see Figure 6.15),which occurs around 10 miles/hour. As traffic builds up pastthis optimum, average speeds show little deterioration, but thereis excessive queuing to get into the network (either from carparking lots within the network or on streets leading into thedesignated network). Godfrey also notes that expanding anintersection to accommodate more traffic will move the queue toanother location within the network, unless the bottlenecksdownstream are cleared.

6.2.3 General Network ModelsIncorporating Network Parameters

Some models have defined specific parameters which intend toquantify the quality of traffic service provided to the users in thenetwork. Two principal models are discussed in this chapter, the

Zahavi (1972a; 1972b) selected three principal variables, I, thetraffic intensity (here defined as the distance traveled per unitarea), R, the road density (the length or area of roads per unitarea), and v, the weighted space mean speed. Using data fromEngland and the United States, values of I, v, and R were foundfor different regions in different cities. In investigating variousrelationships between I and v/R, a linear fit was found betweenthe logarithms of the variables:

where � and m are parameters. Trends for London andPittsburgh are shown in Figure 6.16. The slope (m) was foundto be close to -1 for all six cities examined, reducing Equation6.27 to

where � is different for each city. Relative values of thevariables were calculated by finding the ratio between observed

��

� � ��

Figure 6.15Relationship Between Average (Journey) Speed of Vehicles

and Total Vehicle Mileage on Network (Godfrey 1969, Figure 2).

Figure 6.14Relationship Between Average (Journey)Speed and Number of Vehicles on TownCenter Network (Godfrey 1969, Figure 1).

��

� � ��

Figure 6.16The ��-Relationship for the Arterial Networks of London and Pittsburgh,

in Absolute Values (Zahavi 1972a, Figure 1).

values of I and v/R for each sector and the average value for theentire city. The relationship between the relative values isshown in Figure 6.17, where the observations for London andPittsburgh fall along the same line.

The physical characteristics of the road network, such as streetwidths and intersection density, were found to have a strongeffect on the value of � for each zone in a city. Thus, � mayserve as a measure of the combined effects of the networkcharacteristics and traffic performance, and can possibly be usedas an indicator for the level of service. The � map of London isshown in Figure 6.18. Zones are shown by the dashed lines,with dotted circles indicating zone centroids. Values of � werecalculated for each zone and contour lines of equal � weredrawn, showing areas of (relatively) good and poor traffic flowconditions. (The quality of traffic service improves withincreasing �.)

Unfortunately, Buckley and Wardrop (1980) have shown that �is strongly related to the space mean speed, and Ardekani(1984), through the use of aerial photographs, has shown that �has a high positive correlation with the network concentration.

The two-fluid model also uses parameters to evaluate the levelof service in a network and is described in Section 6.3.

6.2.4 Continuum Models

Models have been developed which assume an arbitrarily finegrid of streets, i.e., infinitely many streets, to circumvent theerrors created on the relatively sparse networks typically usedduring the trip or network assignment phase in transportationplanning (Newell 1980). A basic street pattern is superimposedover this continuum of streets to restrict travel to appropriatedirections. Thus, if a square grid were used, travel on the streetnetwork would be limited to the two available directions (the xand y directions in a Cartesian plot), but origins and destinationscould be located anywhere in the network.

Individual street characteristics do not have to be specificallymodeled, but network-wide travel time averages and capacities(per unit area) must be used for traffic on the local streets. Otherstreet patterns include radial-ring and other grids (triangular, forexample).

��

� � ��

Figure 6.17The ��-Relationship for the Arterial Networks of London and Pittsburgh,

in Relative Values (Zahavi 1972a, Figure 2).

Figure 6.18The ��-Map for London, in Relative Values (Zahavi 1972b, Figure 1).

Vr � Vm frn ,

V � Vm frn�1 .

V � Vm (1 � fs)n�1 .

fs �Ts

T.

��

� � ��

While the continuum comprises the local streets, the major (within the constraints provided by the superimposed grid) to thestreets (such as arterials and freeways) are modeled directly. network of major streets.Thus, the continuum of local streets provides direct access

6.3 Two-Fluid Theory

An important result from Prigogine and Herman's (1971) kinetictheory of traffic flow is that two distinct flow regimes can beshown. These are individual and collective flows and are afunction of the vehicle concentration. When the concentrationrises so that the traffic is in the collective flow regime, the flowpattern becomes largely independent of the will of individualdrivers.

Because the kinetic theory deals with multi-lane traffic, the two-fluid theory of town traffic was proposed by Herman andPrigogine (Herman and Prigogine 1979; Herman and Ardekani1984) as a description of traffic in the collective flow regime inan urban street network. Vehicles in the traffic stream aredivided into two classes (thus, two fluid): moving and stoppedvehicles. Those in the latter class include vehicles stopped in thetraffic stream, i.e., stopped for traffic signals and stop signs,stopped for vehicles loading and unloading which are blockinga moving lane, stopped for normal congestion, etc., but excludesthose out of the traffic stream (e.g., parked cars).

The two-fluid model provides a macroscopic measure of thequality of traffic service in a street network which is independentof concentration. The model is based on two assumptions:

(1) The average running speed in a street network isproportional to the fraction of vehicles that are moving,and

(2) The fractional stop time of a test vehicle circulating ina network is equal to the average fraction of thevehicles stopped during the same period.

The variables used in the two-fluid model represent network-wide averages taken over a given period of time.

The first assumption of the two-fluid theory relates the averagespeed of the moving (running) vehicles, V , to the fraction ofr

moving vehicles, f , in the following manner:r

(6.29)

where V and n are parameters. V is the average maximumm m

running speed, and n is an indicator of the quality of trafficservice in the network; both are discussed below. The averagespeed, V, can be defined as V f , and combining with Equationr r

6.29,

(6.30)

Since f + f = 1, where f is the fraction of vehicles stopped,r s sEquation 6.30 can be rewritten

(6.31)

Boundary conditions are satisfied with this relation: when f =0,s

V=V , and when f =1, V=0.m s

This relation can also be expressed in average travel times ratherthan average speeds. Note that T represents the average traveltime, T the running (moving) time, and T the stop time, all perr s

unit distance, and that T=1/V, T =1/V , and T =1/V , where Tr r m m mis the average minimum trip time per unit distance.

The second assumption of the two-fluid model relates thefraction of time a test vehicle circulating in a network is stoppedto the average fraction of vehicles stopped during the sameperiod, or

(6.32)

This relation has been proven analytically (Ardekani andHerman 1987), and represents the ergodic principle embeddedin the model, i.e., that the network conditions can be representedby a single vehicle appropriately sampling the network.

Restating Equation 6.31 in terms of travel time,

T � Tm ( 1 � fs)�(n�1) .

T � Tm 1 � (Ts /T ) �(n�1) ,

Tr � Tm

1n�1 T

nn�1 .

Ts � T � Tm

1n�1 T

nn�1 .

ln Tr �

1n�1

ln Tm �

nn�1

ln T

��

� � ��

(6.33)

Incorporating Equation 6.32,

(6.34)

realizing that T = T + T , and solving for T ,r s r

(6.35)

The formal two-fluid model formulation, then, is

(6.36)

A number of field studies have borne out the two-fluid model(Herman and Ardekani 1984; Ardekani and Herman 1987; the y-intercept (i.e., T at T =0), and n by the slope of the curve.Ardekani et al. 1985); and have indicated that urban street Data points representing higher concentration levels lie highernetworks can be characterized by the two model parameters, nand T . These parameters have been estimated usingm

observations of stopped and moving times gathered in eachnetwork. The log transform of Equation 6.35,

(6.37)

provides a linear expression for the use of least squares analysis.

Empirical information has been collected with chase carsfollowing randomly selected cars in designated networks. Runshave been broken into one- or two-mile trips, and the runningtime (T ) and total trip time (T) for each one- or two-mile tripr

from the observations for the parameter estimation. Results tendto form a nearly linear relationship when trip time is plottedagainst stop time (Equation 6.36) as shown in Figure 6.19 fordata collected in Austin, Texas. The value of T is reflected bym

s

along the curve.

Note: Each point represents one test run approximately 1 or 2 miles long.

Figure 6.19Trip Time vs. Stop Time for the Non-Freeway Street Network of the Austin CBD

(Herman and Ardekani 1984, Figure 3).

��

� � ��

6.3.1 Two-Fluid Parameters

The parameter T is the average minimum trip time per unitmdistance, and it represents the trip time that might beexperienced by an individual vehicle alone in the network withno stops. This parameter is unlikely to be measured directly,since a lone vehicle driving though the network very late at nightis likely to have to stop at a red traffic signal or a stop sign.T , then, is a measure of the uncongested speed, and a highermvalue would indicate a lower speed, typically resulting in pooreroperation. T has been found to range from 1.5 to 3.0mminutes/mile, with smaller values typically representing betteroperating conditions in the network.

As stop time per unit distance ( T ) increases for a single values

of n, the total trip time also increases. Because T=T +T , ther s total trip time must increase at least as fast as the stop time. Ifn=0, T is constant (by Equation 6.35), and trip time wouldr

increase at the same rate as the stop time. If n>0, trip timeincreases at a faster rate than the stop time, meaning that runningtime is also increasing. Intuitively, n must be greater than zero,since the usual cause for increased stop time is increased

congestion, and when congestion is high, vehicles when moving,travel at a lower speed (or higher running time per unit distance)than they do when congestion is low. In fact, field studies haveshown that n varies from 0.8 to 3.0, with a smaller valuetypically indicating better operating conditions in the network.In other words, n is a measure of the resistance of the network todegraded operation with increased demand. Higher values of nindicate networks that degrade faster as demand increases.Because the two-fluid parameters reflect how the networkresponds to changes in demand, they must be measured andevaluated in a network over the entire range of demandconditions.

While lower n and T values represent, in general, better trafficmoperations in a network, often there is a tradeoff. For example,two-fluid trends for four cities are shown in Figure 6.20. Incomparing Houston (T =2.70 min/mile, n=0.80) and Austinm

(T =1.78 min/mile, n=1.65), one finds that traffic in Austinmmoves at significantly higher average speeds during off-peakconditions (lower concentration); at higher concentrations, thecurves essentially overlap, indicating similar operatingconditions. Thus, despite a higher value of n, traffic conditions

Note: Trip Time vs. Stop Time Two-Fluid Model Trends for CBD Data From the Cities of Austin, Houston, and San Antonio,Texas, and Matamoros, Mexico.

Figure 6.20Trip Time vs. Stop Time Two-Fluid Model Trends


��

� � ��

are better in Austin than Houston, at least at lower being chased imitating the other driver's actions so as to reflect,concentrations. Different values of the two-fluid parameters are as closely as possible, the fraction of time the other driver spendsfound for different city street networks, as was shown above and stopped. The objective is to sample the behavior of the driversin Figure 6.21. The identification of specific features which have in the network as well as the commonly used routes in the streetthe greatest effect on these parameters has been approached network. The chase car's trip history is then broken into one-through extensive field studies and computer simulation.

6.3.2 Two-Fluid Parameters: Influence of Driver Behavior

Data for the estimation of the two-fluid parameters is collectedthrough chase car studies, where the driver is instructed to followa randomly selected vehicle until it either parks or leaves thedesignated network, after which a nearby vehicle is selected andfollowed. The chase car driver is instructed to follow the vehicle

mile (typically) segments, and T and T calculated for each mile.r

The (T ,T) observations are then used in the estimation of thertwo-fluid parameters.

One important aspect of the chase car study is driver behavior,both that of the test car driver and the drivers sampled in thenetwork. One study addressed the question of extreme driverbehaviors, and found that a test car driver instructed to driveaggressively established a significantly different two-fluid trendthan one instructed to drive conservatively in the same networkat the same time (Herman et al. 1988).

Note: Trip Time vs. Stop Time Two-Fluid Model Trends for Dallas and Houston, Texas, compared to the trends in Milwaukee,Wisconsin, and in London and Brussels.

Figure 6.21Trip Time vs. Stop Time Two-Fluid Model Trends Comparison


��

� � ��

The two-fluid trends resulting from the these studies in two cities at lower network concentrations, the aggressive driver can takeare shown in Figure 6.22. In both cases, the normal trend was advantage of the less crowded streets and significantly lower hisfound through a standard chase car study, conducted at the same trip times.time as the aggressive and conservative test drivers were in thenetwork. In both cases, the two-fluid trends established by the As shown in Figure 6.22b, aggressive driving behavior moreaggressive and conservative driver are significantly different. In closely reflects normal driving habits in Austin, suggesting moreRoanoke (Figure 6.22a), the normal trend lies between the aggressive driving overall. Also, all three trends converge ataggressive and conservative trends, as expected. However, the high demand (concentration) levels, indicating that, perhaps, theaggressive trend approaches the normal trend at high demand Austin network would suffer congestion to a greater extent thanlevels, reflecting the inability of the aggressive driver to reduce Roanoke, reducing all drivers to conservative behavior (at leasthis trip and stop times during peak periods. On the other hand, as represented in the two-fluid parameters).

Note: The two-fluid trends for aggressive, normal, and conservative drivers in (a) Roanoke, Virginia, and (b) Austin, Texas

Figure 6.22Two-Fluid Trends for Aggressive, Normal, and Conservative Drivers

(Herman et al. 1988, Figures 5 and 8).

Tm � 3.59 � 0.54 C6 andn � �0.21 � 2.97 C3 � 0.22 C7

Tm � 3.93 � 0.0035 X5 � 0.047 X6 � 0.433 X10and n � 1.73 � 1.124 X2 � 0.180 X3 � 0.0042 X5 � 0.271 X9

��

� � ��

(6.38)

(6.39)

The results of this study reveal the importance of the behavior of approaches with signal progression. Of these, only two featuresthe chase car driver in standard two-fluid studies. While the (average block length and intersection density) can beeffects on the two-fluid parameters of using two different chase considered fixed, and, as such, not useful in formulating networkcar drivers in the same network at the same time has not been improvements. In addition, one feature (average number ofinvestigated, there is thought to be little difference between two lanes per street), also used in the previous study (Ayadh 1986),well-trained drivers. To the extent possible, however, the same can typically be increased only by eliminating parking (ifdriver has been used in different studies that are directly present), yielding only limited opportunities for improvement ofcompared. traffic flow. Data was collected in ten cities; in seven of the

6.3.3 Two-Fluid Parameters: Influenceof Network Features (Field Studies)

Geometric and traffic control features of a street network alsoplay an important role in the quality of service provided by anetwork. If relationships between specific features and the two-fluid parameters can be established, the information could beused to identify specific measures to improve traffic flow andprovide a means to compare the relative improvements.

Ayadh (1986) selected seven network features: lane miles persquare mile, number of intersections per square mile, fraction ofone-way streets, average signal cycle length, average blocklength, average number of lanes per street, and average blocklength to block width ratio. The area of the street network underconsideration is used with the first two variables to allow a directcomparison between cities. Data for the seven variables werecollected for four cities from maps and in the field. Through aregression analysis, the following models were selected:

where C is the fraction of one-way streets, C the average3 6

number of lanes per street, and C the average block length to7block width ratio. Of these network features, only one (thefraction of one-way streets) is relatively inexpensive toimplement. One feature, the block length to block width ratio, by Computer Simulationis a topological feature which would be considered fixed for anyestablished street network.

Ardekani et al. (1992), selected ten network features: averageblock length, fraction of one-way streets, average number oflanes per street, intersection density, signal density, averagespeed limit, average cycle length, fraction of curb miles withparking allowed, fraction of signals actuated, and fraction of

cities, more than one study was conducted as major geometricchanges or revised signal timings were implemented, yieldingnineteen networks for this study. As before, the two-fluidparameters in each network were estimated from chase car dataand the network features were determined from maps, fieldstudies, and local traffic engineers. Regression analysis yieldedthe following models:

where X is the fraction of one-way streets, X the average2 3

number of lanes per street, X the signal density, X the average5 6

speed limit, X the fraction of actuated signals, and X the9 10fraction of approaches with good progression. The R for these2

equations, 0.72 and 0.75 (respectively), are lower than those forEquation 6.38 (both very close to 1), reflecting the larger datasize. The only feature in common with the previous model(Equation 6.38) is the appearance of the fraction of one-waystreets in the model for n. Since all features selected can bechanged through operational practices (signal density can bechanged by placing signals on flash), the models have potentialpractical application. Computer simulation has also been usedto investigate these relationships, and is discussed in Section6.3.4.

6.3.4 Two-Fluid Parameters: Estimation

Computer simulation has many advantages over field data in thestudy of network models. Conditions not found in the field canbe evaluated and new control strategies can be easily tested. Inthe case of the two-fluid model, the entire vehicle population inthe network can be used in the estimation of the modelparameters, rather than the small sample used in the chase carstudies. TRAF-NETSIM (Mahmassani et al. 1984), a

��

� � ��

microscopic traffic simulation model, has been used successfully can be simulated by recording the trip history of a single vehiclewith the two-fluid model. for one mile, then randomly selecting another vehicle in the

Most of the simulation work to-date has used a generic grid (specifically, Equation 6.35, the log transform of which is usednetwork in order to isolate the effects of specific network to estimate the parameters), estimations performed at thefeatures on the two-fluid parameters (FHWA 1993). Typically, network level and at the individual vehicle level result inthe simulated network has been a 5 x 5 intersection grid made up different values of the parameters, and are not directlyentire of two-way streets. Traffic signals are at each intersection comparable. The sampling strategy, which was found to provideand uniform turning movements are applied throughout. The the best parameter estimates, required a single vehiclenetwork is closed, i.e., vehicles are not allowed to leave the circulating in the network for at least 15 minutes. However, duenetwork, thus maintaining constant concentration during the to the wide variance of the estimate (due to the possibility of asimulation run. The trip histories of all the vehicles circulating relatively small number of "chased" cars dominating the samplein the network are aggregated to form a single (T , T)robservation for use in the two-fluid parameter estimation. Aseries of five to ten runs over a range of network concentrations(nearly zero to 60 or 80 vehicles/lane-mile) are required toestimate the two-fluid parameters.

Initial simulation runs in the test network showed both T and Ts

increasing with concentration, but T remaining nearly constant,r

indicating a very low value of n (Mahmassani et al. 1984). Inits default condition, NETSIM generates few of the vehicleinteraction of the type found in most urban street networks,resulting in flow which is much more idealized than in the field.The short-term event feature of NETSIM was used to increasethe inter-vehicular interaction (Williams et al. 1985). With thisfeature, NETSIM blocks the right lane of the specified link atmid-block; the user specifies the average time for each blockageand the number of blockages per hour, which are stochasticallyapplied by NETSIM. In effect, this represents a vehicle stoppingfor a short time (e.g., a commercial vehicle unloading goods),blocking the right lane, and requiring vehicles to change lanes togo around it. The two-fluid parameters (and n in particular)were very sensitive to the duration and frequency of the short-term events. For example, using an average 45-second eventevery two minutes, n rose from 0.076 to 0.845 and T fell fromm2.238 to 2.135. With the use of the short-term events, the valuesof both parameters were within the ranges found in the fieldstudies. Further simulation studies found both block length(here, distance between signalized intersections) and the use ofprogression to have significant effects on the two-fluidparameters (Williams et al. 1985).

Simulation has also provided the means to investigate the use ofthe chase car technique in estimating the two-fluid parameters(Williams et al. 1995). The network-wide averages in asimulation model can be directly computed; and chase car data

network. Because the two-fluid model is non-linear

estimation), the estimate using a single vehicle was often farfrom the parameter estimated at the network level. On the otherhand, using 20 vehicles to sample the network resulted inestimates much closer to those at the network level. The muchsmaller variance of the estimates made with twenty vehicles,however, resulted in the estimate being significantly differentfrom the network-level estimate. The implication of this studyis that, while estimates at the network and individual vehiclelevels can not be directly compared, as long as the samesampling strategy is used, the resulting two-fluid parameters,although biased from the "true" value, can be used in makingdirect comparisons.

6.3.5 Two-Fluid Parameters: Influence of Network Features(Simulation Studies)

The question in Section 6.3.3, above, regarding the influence ofgeometric and control features of a network on the two-fluidparameters was revisited with an extensive simulation study(Bhat 1994). The network features selected were: averageblock length, fraction of one-way streets, average number oflanes per street, signals per intersection, average speed limit,average signal cycle length, fraction of curb miles with parking,and fraction of signalized approaches in progression. Auniform-precision central composite design was selected as theexperimental design, resulting in 164 combination of the eightnetwork variables. The simulated network was increased to 11by 11 intersections; again, vehicles were not allowed to leave thenetwork, but traffic data was collected only on the interior 9 by9 intersection grid, thus eliminating the edge effects caused bythe necessarily different turning movements at the boundaries.Ten simulation runs were made for each combination of

Tm � 1.049 � 1.453 X2 � 0.684 X3 � 0.024 X6 andn � 4.468 � 1.391 X3 � 0.048 X5 � 0.042 X6

V � f (K) ,

��

� � ��

(6.40)

(6.41)

variables over a range of concentrations from near zero to about NETSIM reflected traffic conditions in San Antonio, NETSIM35 vehicles/lane-mile. was calibrated with the two-fluid model.

Regression analysis yielded the following models: Turning movement counts used in the development of the new

where X is the fraction of one-way streets, X the number of2 3

lanes per street, X average speed limit, and X average cycle5 6length. The R (0.26 and 0.16 for Equation 6.40) was2

considerably lower than that for the models estimated with datafrom field studies (Equation 6.39). Additionally, the onlyvariable in common between Equations 6.39 and 6.40 is thenumber of lanes per street in the equation for n. Additional workis required to clarify these relationships.

6.3.6 Two-Fluid Model:A Practical Application

When the traffic signals in downtown San Antonio were retimed,TRAF-NETSIM was selected to quantify the improvements inthe network. In order to assure that the results reported by

signal timing plans were available for coding NETSIM.Simulation runs were made for 31 periods throughout the day,and the two-fluid parameters were estimated and compared withthose found in the field. By a trial and error process, NETSIMwas calibrated by

� Increasing the sluggishness of drivers, by increasingheadways during queue discharge at traffic signalsand reducing maximum acceleration,

� Adding vehicle/driver types to increase the range ofsluggishness represented in the network, and

� Reducing the desired speed on all links to 32.2 km/hduring peaks and 40.25 km/h otherwise (Denney1993).

Three measures of effectiveness (MOEs) were used in theevaluation: total delay, number of stops, and fuel consumption.The changes noted for all three MOEs were greater betweencalibrated and uncalibrated NETSIM results than betweenbefore and after results. Reported relative improvements werealso affected. The errors in the reported improvements withoutcalibration ranged from 16 percent to 132 percent (Denney1994).

6.4 Two-Fluid Model and Traffic Network Flow Models

Computer simulation provides an opportunity to investigate fluid model, to the concentration. In addition, using values ofnetwork-level relationships between the three fundamental flow, speed, and concentration independently computed from thevariables of traffic flow, speed (V), flow (Q), and concentration(K), defined as average quantities taken over all vehicles in thenetwork over some observation period (Mahmassani et al.1984). While the existence of "nice" relations between thesevariables could not be expected, given the complexity of networkinterconnections, simulation results indicate relationships similarto those developed for arterials may be appropriate (Mahmassani model system assumed Q=KV and the two-fluid model, andet al. 1984; Williams et al. 1985). A series of simulation runs, consisted of three relations:as described in Section 6.3.4, above, was made at concentrationlevels between 10 and 100 vehicles/lane-mile. The results areshown in Figure 6.23, and bear a close resemblance to theircounterparts for individual road sections. The fourth plot showsthe relation of f , the fraction of vehicles stopped from the two-s

simulations, the network-level version of the fundamentalrelation Q=KV was numerically verified (Mahmassani et al.1984; Williams et al. 1987).

Three model systems were derived and tested against simulationresults (Williams et al. 1987; Mahmassani et al. 1987); each

fs � h (K) .

Q � g (K) , and

��

� � ��

(6.43)

(6.42)

Figure 6.23Simulation Results in a Closed CBD-Type Street Network.

(Williams et al. 1987, Figures 1-4).

A model system is defined by specifying one of the above relationships; the other two can then be analytically derived. (Arelation between Q and V could also be derived.)

Model System 1 is based on a postulated relationship betweenthe average fraction of vehicles stopped and the networkconcentration from the two-fluid theory (Herman and Prigogine

fs � fs,min � ( 1�fs,min ) ( K/Kj )� ,

V � Vm (1�fs,min )n�1 [1�(K/Kj )�]n�1 ,

Q � K Vm (1�fs,min )n�1 [1�(K/Kj )�]n�1 .

V � Vf (1�K/Kj ) ,

fs � 1 � [(Vf /Vm) (1�K/Kj) ]1/(n�1) ,

Q � Vf (K� K 2 /Kj ) .

V � Vf exp[��(K/Km )d ] ,

fs�1�{(Vf /Vm ) exp [��(K/Km )d ]}1/(n�1) and

Q � K Vf exp[��(K/Km )d ] .

��

� � ��

(6.44)

(6.45)

(6.46)

(6.47)

(6.48)

(6.49)

(6.50)

(6.51)

(6.52)

1979), later modified to reflect that the minimum f > 0 then by using Q=KV,s(Ardekani and Herman 1987):

where f is the minimum fraction of vehicles stopped in as,min

network, K is the jam concentration (at which the network isj

effectively saturated), and � is a parameter which reflects thequality of service in a network. The other two relations can bereadily found, first by substituting f from Equation 6.44 intosEquation 6.31:

then by using Q=KV,

Equations 6.44 through 6.46 were fitted to the simulated dataand are shown in Figure 6.24. Because the point representingthe highest concentration (about 100 vehicles/lane-mile) did notlie in the same linear lnT - lnT trend as the other points, the two-r

fluid parameters n and T were estimated with and without themhighest concentration point, resulting in the Method 1 andMethod 2 curves, respectively, in the V-K and Q-K curves inFigure 6.24.

Model System 2 adopts Greenshields' linear speed-concentrationrelationship (Gerlough and Huber 1975),

where V is the free flow speed (and is distinct from V ;f m

V � V always, and typically V < V ). The f -K relation can bef m f m sfound by substituting Equation 6.47 into Equation 6.31 andsolving for f :s

Equations 6.47 through 6.49 were fitted to the simulation dataand are shown in Figure 6.25. The difference between theMethod 1 and Method 2 curves in the f -K plot (Figure 6.25) issdescribed above. Model System 3 uses a non-linear bell-shapedfunction for the V-K model, originally proposed by Drake, et al.,for arterials (Gerlough and Huber 1975):

where K is the concentration at maximum flow, and � and d arem

parameters. The f -K and Q-K relations can be derived as shownsfor Model System 2:

Equations 6.50 through 6.52 were fitted to the simulation dataand are shown in Figure 6.26.

Two important conclusions can be drawn from this work. First,that relatively simple macroscopic relations between network-level variables appear to work. Further, two of the modelsshown are similar to those established at the individual facilitylevel. Second, the two-fluid model serves well as the theoreticallink between the postulated and derived functions, providinganother demonstration of the model's validity. In the second andthird model systems particularly, the derived f -K functionsperformed remarkable well against the simulated data, eventhough it was not directly calibrated using that data.

��

� � ��

<

Figure 6.24Comparison of Model System 1 with Observed Simulation Results

(Williams et al. 1987, Figure 5, 7, and 8).

��

� � ��



��

� � ��

6.5 Concluding Remarks

As the scope of traffic control possibilities widens with the optimization of the control system) becomes clear. While thedevelopment of ITS (Intelligent Transportation Systems) models discussed in this chapter are not ready for easyapplications, the need for a comprehensive, network-wide implementation, they do have promise, as in the application ofevaluation tool (as well as one that would assist in the the two-fluid model in San Antonio (Denney et al. 1993; 1994).

References

Angel, S. and G. M. Hyman (1970). Urban Velocity Fields. Environment and Planning, Vol. 2.

Ardekani, S. A. (1984). The Two-Fluid Characterization ofUrban Traffic: Theory, Observation, and Experiment. Conference), S. Yagar, A. Santiago, editors, FederalPh.D. Dissertation, University of Texas at Austin. Highway Administration, U.S. Department of

Ardekani, S. A., V. Torres-Verdin, and R. Herman (1985). Transportation.The Two-Fluid Model and Traffic Quality in Mexico City(El Modelo Bifluido y la Calidad del Tránsito en la Ciudadde México). Revista Ingeniería Civil.

Ardekani, S. A. and R. Herman, (1987). Urban Network-WideVariables and Their Relations. Transportation Science,Vol. 21, No. 1.

Ardekani, S. A., J. C. Williams,, and S. Bhat, (1992). Influenceof Urban Network Features on Quality of Traffic Service.Transportation Research Record 1358, TransportationResearch Board.

Ayadh, M. T. (1986). Influence of the City GeometricFeatures on the Two Fluid Model Parameters, M.S.Thesis, Virginia Polytechnic Institute and State University.

Beimborn, E. A. (1970). A Grid Travel Time Model.Transportation Science, Vol. 4.Bhat, S .C. S. (1994). Effects of Geometric and Control

Features on Network Traffic: A Simulation Study. Ph.D.Dissertation, University of Texas at Arlington.

Branston, D. M. (1974). Urban Traffic Speeds—I: AComparison of Proposed Expressions Relating JourneySpeed to Distance from a Town Center. TransportationScience, Vol. 8, No. 1.

Buckley, D. G. and J. G. Wardrop, (1980). Some GeneralProperties of a Traffic Network. Australian RoadResearch, Vol. 10, No. 1.

Denney, Jr., R. W., J. C. Williams, S. C. S. Bhat, and S. A. Ardekani, (1993). Calibrating NETSIM for a CBDUsing the Two-Fluid Model. Large Urban Systems(Proceedings of the Advanced Traffic Management

Denney, Jr., R. W., J. C. Williams, and S. C. S. Bhat, (1994).Calibrating NETSIM Using the Two-Fluid Model.Compendium of Technical Papers (Proceedings of the 64thITE Annual Meeting, Dallas), Institute of TransportationEngineers.

Federal Highway Administration, U. S. Department ofTransportation (1993). TRAF User Reference Guide,Version 4.0.

Gerlough, D. L. and M. J. Huber, (1975). Traffic FlowTheory: A Monograph, Special Report 165, TransportationResearch Board.

Godfrey, J. W. (1969). The Mechanism of a Road Network.Traffic Engineering and Control, Vol. 11, No. 7.

Herman, R. and S. A. Ardekani, (1984).Characterizing Traffic Conditions in Urban Areas.Transportation Science, Vol. 18, No. 2.

Herman, R. and I. Prigogine, (1979). A Two-Fluid Approach toTown Traffic. Science, Vol. 204, pp. 148-151.

Hutchinson, T. P. (1974). Urban Traffic Speeds—II: Relationof the Parameters of Two Simpler Models to Size of theCity and Time of Day. Transportation Science, Vol. 8, No. 1.

Lyman, D. A. and P. F. Everall, (1971). Car Journey Times inLondon—1970. RRL Report LR 416, Road ResearchLaboratory.

Mahmassani, H., J. C. Williams, and R. Herman, (1984).Investigation of Network-Level Traffic Flow Relationships:Some Simulation Results. Transportation Research Record971, Transportation Research Board.

��

� � ��

Mahmassani, H. S., J. C. Williams, and R. Herman, (1987). Performance of Urban Traffic Networks. Transportationand Traffic Theory (Proceedings of the Tenth Internationalon Transportation and Traffic Theory Symposium,Cambridge, Massachusetts), N.H. Gartner, N.H.M. Wilson,editors, Elsevier.

Newell, G. F. (1980). Traffic Flow on TransportationNetworks, Chapter 4, MIT Press.

Prigogine, I. and R. Herman, (1971). Kinetic Theory ofVehicular Traffic, American Elsevier.

Road Research Laboratory (1965). Research on Road Traffic.Smeed, R. J. (1963). International Road Safety and TrafficReview, Vol. 11, No. 1.Smeed, R. J. and J. G. Wardrop, (1964). An Exploratory

Comparison of the Advantages of Cars and Buses for Williams, J. C., H. S. Mahmassani, and R. Herman (1995).Travel in Urban Areas. Journal of the Institute of Sampling Strategies for Two-Fluid Model ParameterTransportation, Vol. 30, No. 9.

Smeed, R. J. (1965). Journal of the Institute of Mathematics Part A, Vol. 29A, No. 3.and its Applications, Vol. 1.

Smeed, R. J. (1966). Road Capacity of City Centers. TrafficEngineering and Control, Vol. 8, No. 7. Model Parameter Sensitivity. Transportation Research

Smeed, R. J. (1968). Traffic Studies and Urban Congestion.Journal of Transport Economics and Policy, Vol. 2,No. 1. Urban Traffic Network Flow Models. Transportation

Thomson, J. M. (1967a). Speeds and Flows of Traffic inCentral London: 1. Sunday Traffic Survey. TrafficEngineering and Control, Vol. 8, No. 11. Networks by the �-Relationship. Traffic Engineering and

Thomson, J. M. (1967b). Speeds and Flows of Trafficin Central London: 2. Speed-Flow Relations. TrafficEngineering and Control, Vol. 8, No. 12. Networks by the �-Relationship, Part 2. Traffic

Transportation Research Board (1994). Highway CapacityManual, Special Report 209, (with revisions).

Vaughan, R., A. Ioannou, and R. Phylactou (1972). TrafficCharacteristics as a Function of the Distance to the TownCentre. Traffic Engineering and Control, Vol. 14, No. 5.

Wardrop, J. G. (1952). Some Theoretical Aspects of RoadTraffic Research. Proceedings of the Institution of CivilEngineers, Vol. 1, Part 2.

Wardrop, J. G. (1968). Journey Speed and Flow in CentralUrban Areas. Traffic Engineering and Control, Vol. 9, No.11.

Wardrop, J. G. (1969). Minimum Cost Paths in Urban Areas.Beiträge zur Theorie des Verkehrsflusses, Universitat (TH)Karlsruhe.

Estimation in Urban Networks. Transportation Research -

Williams, J. C., H. S. Mahmassani, and R. Herman, (1985). Analysis of Traffic Network Flow Relations and Two-Fluid

Record 1005, Transportation Research Board.Williams, J. C., H. S. Mahmassani, and R. Herman, (1987).

Research Record 1112, Transportation Research Board.Zahavi, Y. (1972a). Traffic Performance Evaluation of Road

Control, Vol. 14, No. 5.Zahavi, Y. (1972b). Traffic Performance Evaluation of Road

Engineering and Control, Vol. 14, No. 6.

Associate Professor, Civil Engineering Department, University of Texas at Arlington, Box 19308,10

Arlington, TX 76019-0308

Professor, Department of Civil Engineering, University of Toronto, Toronto, Ontario Cananda M5S 1A411

Transportation Planning Engineer, Virginia Department of Transportation, Fairfax, Va.12

TRAFFIC IMPACT MODELS

BY SIA ARDEKANI 10

EZRA HAUER11

BAHRAM JAMEI12

Chapter 7 - Frequently used Symbols

� = fuel consumption per unit distance� = parametera = instantaneous acceleration rate (a>0, km/hr/sec)

A = acceleration rate AADT = average annual daily traffic (vehicles/day) ACCT = acceleration timeADT = average daily traffic (vehicles/day)C1 = average peak hour concentrationC8 = annual maximum 8-hour concentrationCLT = Deceleration timed = average stopped delay per vehicle (secs)s

E = Engine size (cm )3

EFI = idle emission factorEFL = intersection link composite emission factorf = instantaneous fuel consumption (mL/sec)F = fuel consumed (lit/km) [Watson, et al model]F = average fuel consumption per roadway section (mL) [Akcelik model]f = fuel consumption rate while cruising (mL/km)1

f = fuel consumption rate while idling (mL/sec)2

f = excess fuel consumption per vehicle stop (mL)3

f = steady-state fuel consumption rate at cruising speed (mL/km)c

h = average number of stops per vehicleK = parameter representing idle flow rate (mL/sec)1

K = parameter representing fuel consumption to overcome rolling resistance2

K ,K = parameters related to fuel consumption due to positive acceleration4 5

L = payload (Kg)LQU = queue length m = expected number of single-vehicle accidents per unit time NDLA = vehicles delayed per cycle per laneP = probability of a single-vehicle accident PKE = positive kinetic energyq = flow (vph)q = flow rate of type i vehicles (vph) iS = speedSPD = cruise speedT = average travel time per unit distanceu = model parameter related to driving conditionsV = average speed (km/hr) [Elemental model]V = instantaneous speed (km/hr) [Akcelik and Bayley model]V = steady-state cruising speed (km/hr)c

V = final speed (km/hr)f

V = initial speed (km/hr)i

VMT = vehicle miles of travelV = space mean speed (km/hr)s

VSP = at-rest vehicle spacing

� � �

7.TRAFFIC IMPACT MODELS

7.1 Traffic and Safety

7.1.1 Introduction

This section ought to be about how traffic flow, speed and the pertains to the same period of time which the accident frequencylike are related to accident frequency and severity. However, represents. Thus, eg., if the ordinate shows the expected numberdue to limitation of space, only the relationship between accident of fatal accidents/year in 1972-1976 for a certain road section,frequency and traffic flow will be discussed. The terminology then the AADT is the average for the period 1972-1976. that pertains to characteristics of the traffic stream has alreadybeen established and only a few definitions need to be added.The ‘safety’ of an entity is defined as ‘the number of accidents Thus, eg., head-on collisions may depend on the two opposingby type, expected to occur on the entity in a certain period,per unit of time’. In this definition, ‘accident types’ arecategories such as rear-end, sideswipe, single-vehicle, multi-vehicle, injury, property damage only, etc. The word ‘expected’is as in probability theory: what would be the average-in-the-long-run if it was possible to freeze all the relevantcircumstances of the period at their average, and then repeat it In practice it is common to use the term ‘accident rate’. Theover and over again. The word ‘entity’ may mean a specific road accident rate is proportional to the slope of the line joining thesection or intersection, a group of horizontal curves with the origin and a point of the safety performance function. Thus, atsame radius, the set of all signalized intersections in point A of Figure 7.1, where AADT is 3000 vehicles per dayPhiladelphia, etc. Since the safety of every entity changes in and where the expected number of accidents for this road sectiontime, one must be specific about the period. Furthermore, to is 1.05 accidents per year, the accident rate isfacilitate communication, safety is usually expressed as a 1.05/(3000×365)=0.96×10 accidents/vehicle. At point B thefrequency. Thus, eg., one might speak about the expected accident rate is 1.2/(4000×365)=0.82×10 accidents/vehicle. Ifnumber of fatal accidents/year in 1972-1976 for a certain road the road section was, say, 1.7 km long, the same accident ratessection. To standardize further one often divides by the section could be written as 1.05/(3000×365×1.7)=0.56×10length; now the units may be, say, accidents/(year × km). accidents/vehicle-km and 1.2/(4000×365×1.7)=0.48×10

As defined, the safety of an entity is a string of expectedfrequencies, m , m , . . . ,m , . . . , one for each accident type1 2 ichosen. However, for the purpose of this discussion it willsuffice to speak about one (unspecified) accident type, theexpected accident frequency of which is m .i

7.1.2 Flow and Safety

The functional relationship between m and the traffic flow whichi

the entity serves, is a ‘safety performance function’. A safetyperformance function is depicted schematically in Figure 7.1.For the moment its shape is immaterial. It tells how for someentity the expected frequency of accidents of some type would bechanging if traffic flow on the entity changed while all otherconditions affecting accident occurrence remained fixed. Whilethe flow may be in any units, it is usually understood that it

Naturally, m can be a function of more than one traffic flow.i

flows; collisions between pedestrians and left-turning trafficdepend on the flow of pedestrians, the flow of straight-throughvehicles, and the flow of left-turning vehicles etc.. In short, thearguments of the safety performance function can be severalflows.

-6

-6

-6

-6

accidents/vehicle-km.

The safety performance function of an entity is seldom a straightline. If so, the accident rate is not constant but varies with trafficflow. As a consequence, if one wishes to compare the safety oftwo or more entities serving different flows, one can not use theaccident rate for this purpose. The widespread habit of usingaccident rates to judge the relative safety of different entities orto assess changes in safety of the same entity is inappropriate andoften harmful. To illustrate, suppose that the AADT on the roadsection in Figure 7.1 increased from 3000 ‘before pavementresurfacing’ to 4000 ‘after pavement resurfacing’ and that theaverage accident frequency increased from 1.05 ‘before’ to 1.3‘after’. Note that 1.2 accidents/year would be expected atAADT=4000 had the road surface remained unchanged (seeFigure 7.1). Since 1.3 > 1.2 one must conclude that followingresurfacing there was a deterioration of 0.1 accidents/year. But

��

� � �

Figure 7.1Safety Performance Function and Accident Rate.

less than the accident rate ‘before resurfacing’ (1.05/3000×365=0.96×10 ) which erroneously suggests that there has been an-6

improvement. Similar arguments against the use of accidentrates can be found in Pfundt (1969), Hakkert et al. (1976),Mahalel (1986), Brundell-Freij & Ekman (1991), Andreassen(1991).

To avoid such errors, the simple rule is that safety comparisonsare legitimate only when the entities or periods are compared asif they served the same amount of traffic. To accomplishsuch equalization, one needs to know the safety performancefunction. Only in the special case when the performancefunction happens to be a straight line, may one divide by trafficflow and then compare accident rates. However, to judgewhether the safety performance function is a straight line, onemust know its shape, and when the shape of the safetyperformance function is known, the computation of an accidentrate is superfluous. It is therefore best not to make use ofaccident rates. For this reason, the rest of the discussion is aboutexpected accident frequencies, not rates.

Knowledge of safety performance functions is an importantelement of rational road safety management. The nature andshape of this function is subject to some logical considerations.However, much of the inquiry must be empirical.

7.1.3 Logical Considerations

It stands to reason that there is some kind of relationshipbetween traffic flow and safety. For one, without traffic there areno traffic accidents. So, the safety performance function must gothrough the origin. Also, the three interrelated characteristics ofthe traffic stream - flow, speed and density - all influence thethree interrelated aspects of safety - the frequency ofopportunities for accidents to occur, the chance of accidentoccurrence given an opportunity, and the severity of the outcomegiven an accident. However, while a relationship may bepresumed to exist, it is rather difficult to learn much about itsmathematical form by purely deductive reasoning.

Using logic only, one could argue as follows: "If, as inprobability theory, the passage of a vehicle through a roadsection or an intersection is a ‘trial’ the ‘outcome’ of which canbe ‘accident’ or ‘no-accident’ with some fixed probability.Assume further that vehicle passages are so infrequent that thisprobability is not influenced by the frequency at which the‘trials’ occur. Under such conditions the expected number ofsingle-vehicle accidents in a fixed time period must beproportional to the number of trials in that time period - that isto flow." In symbols, m =qp, where q is flow and p is thesingle-vehicleprobability of a single-vehicle accident in one passage of a

��

� � �

vehicle. In this, p is a constant that does not depend on q. Thus,one may argue, that the number of single-vehicle accidents oughtto be proportional to flow, but only at very low flows.

As flow and density increase to a point where a driver can seethe vehicle ahead, the correspondence between the mentalpicture of independent trials and between reality becomesstrained. The probability of a ‘trial’ to result in a single-vehicleaccident now depends on how close other vehicles are; that isp = p(q). Should p(q) be an increasing function of q, thenm would increase more than in proportion with flow.single-vehicle Conversely, if an increase in flow diminishes the probability thata vehicle passing the road section will be in a single-vehicleaccident, then m would increase less than in proportionsingle-vehicle

to traffic flow; indeed, m = qp(q) can even decrease assingle-vehicletraffic flow increases beyond a certain point. Thus, by logicalreasoning one can only conclude that near the origin, the safetyperformance function for single-vehicle accidents ought to be astraight line.

If the safety performance function depends on two conflictingflows (car-train collisions at rail-highway grade crossings, car-truck collisions on roads, car-pedestrian collision at intersectionsetc.) then, near the origin, m should be proportional to thei

product of the two flows. One could also use the paradigm ofprobability theory to speculate that (at very low flows) theexpected number of collisions with vehicles parked on shouldersis proportional to the square of the flows: in the language of‘trials’, ‘outcomes’, the number of vehicles parked on theshoulder ought to be proportional to the passing flow and thenumber of vehicles colliding with the parked cars ought to beproportional to the same flow. From here there is only a smallstep to argue that, say, the number of rear-end collisions shouldalso be proportional to q . Again, this reasoning applies only to2

very low flows. How m depends on q when speed choice,alertness and other aspects of behavior are also a function offlow, cannot be anticipated by speculation alone.

This is as far as logical reasoning seems to go at present. It onlytells us what the shape of the safety performance function shouldbe near the origin. Further from the origin, when p changes withq, not much is gained thinking of m as the product qp(q). Sincethe familiar paradigm of ‘trials’ and ‘outcomes’ ceased to fitreality, and the notion of ‘opportunity to have an accident’ isvague, it might be better to focus directly on the function m =m(q) instead of its decomposition into the product qp(q).

Most theoretical inquiry into the relationship between flow andsafety seems to lack detail. Thus, eg., most researchers try torelate the frequency of right-angle collisions at signalizedintersections to the two conflicting flows. However, onreflection, the second and subsequent vehicles of a platoon mayhave a much lesser chance to be involved in such a collision thanthe first vehicle. Therefore it might make only a slight differencewhether 2 or 20 vehicles have to stop for the same red signal.For this reason, the total flow is likely to be only weakly andcircuitously related to the number of situations which generateright angle collisions at signalized intersections. There seems tobe scope and promise for more detailed, elaborate and realistictheorizing. In addition, most theorizing to date attempted torelate safety to flow only. However, since flow, speed anddensity are connected, safety models could be richer if theycontained all relevant characteristics of the traffic stream. Thus,eg., a close correspondence has been established between thenumber of potential overtakings derived from flow and speeddistribution and accident involvement as a function of speed(Hauer 1971). Welbourne (1979) extends the ideas to crossingtraffic and collisions with fixed objects. Ceder (1982) attemptsto link accidents to headway distributions (that are a function offlow) through probabilistic modeling.

There is an additional aspect of the safety performance functionwhich may benefit from logical examination. The claim was thatit is reasonable to postulate the existence of a relationshipbetween the traffic triad ‘flow, speed and density’ and betweenthe safety triad ‘frequency of opportunities, chance of accidentgiven opportunity, and severity of the outcome given accident’.However, if there is a cause-effect relationship, it must bebetween accidents and the traffic characteristics near the time oftheir occurrence. One must ask whether there still is somemeaningful safety performance function between accidents andtraffic flow when flow is averaged over, say, a year. Whether thehabit of relating accidents to AADTs (that is, averages over ayear) materially distorts the estimated safety performancefunction is at present unclear. In a study by Quaye et al. (1993)three separate models were estimated from 15 minute flows,which then were aggregated into 1 hour flows and then into 7hour flows. The three models differed but little. Persaud andDzbik (1993) call models that relate hourly flows to accidents"microscopic" and models that relate AADT to yearly accidentcounts "macroscopic".

As will become evident shortly, empirical inquiries about safetyperformance functions display a disconcerting variety of results.

��

� � �

A part of this variety could be explained by the fact that the most whether the result of a cross section study enables one toubiquitous data for such inquiries consist of flow estimates anticipate how safety of a certain facility would change as awhich are averages pertaining to periods of one year or longer. result of a change in traffic flow.

7.1.4 Empirical Studies

Empirical studies about the association of traffic flow andaccident frequency seldom involve experimentation; their natureis that of fitting functions to data. What is known about safetyperformance functions comes from studies in which theresearcher assembles a set of data of traffic flows and accidentcounts, chooses a certain function that is thought to represent therelationship between the two, and then uses statistical techniquesto estimate the parameters of the chosen function. Accordingly,discussion here can be divided into sections dealing with: (1) thekinds of study and data, (2) functional forms or models, (3)parameter estimates.

7.1.4.1 Kinds Of Study And Data

Data for the empirical investigations are accident counts andtraffic flow estimates. A number-pair consisting of the accidentcount for a certain period and the estimated flow for the sameperiod is a ‘data point’ in a Cartesian representation such asFigure 7.1. To examine the relationship between traffic flow andaccident frequency, many such points covering an adequaterange of traffic flows are required.

There are two study prototypes (for a discussion see eg., Jovanisand Chang 1987). The most common way to obtain data pointsfor a range of flows is to choose many road sections orintersections that are similar, except that they serve differentflows. In this case we have a ‘cross-section’ study. In such astudy the accident counts will reflect not only the influence oftraffic flow but also of all else that goes with traffic flow. Inparticular, facilities which carry larger flows tend to be built andmaintained to higher standards and tend to have better markingsand traffic control features. This introduces a systematic biasinto ‘cross-section’ models. If a road that is built and maintainedto a higher standard is safer, then accident counts on high-flowroads will tend to be smaller than had the roads been built,maintained and operated in the same way as the lower flowroads. Thus, in a cross-section study, it is difficult to separatewhat is due to traffic flow, and what is due to all other factorswhich depend on traffic flow. It is therefore questionable

Less common are studies that relate different traffic flows on thesame facility to the corresponding accident counts. In this casewe have a ’time-sequence’ study. In such a study one obtainsthe flow that prevailed at the time of each accident and thenumber of hours in the study period when various flowsprevailed. The number of accidents in a flow group divided bythe number of hours when such flows prevailed is the accidentfrequency (see, eg., Leutzbach et al. 1970, Ceder and Livneh1978, Hall and Pendelton 1991). This approach might obviatesome of the problems that beset the cross-section study.However, the time-sequence study comes with its owndifficulties. If data points are AADTs and annual accidentcounts over a period of many years, then the range of the AADTsis usually too small to say much about any relationship. Inaddition, over the many years, driver demography, norms ofbehavior, vehicle fleet, weather, and many other factors alsochange. It is therefore difficult to distinguish between what isdue to changes in traffic flow and what is due to the many otherfactors that have also changed. If the data points are traffic flowsand accidents over a day, different difficulties arise. For one, thecount of accidents (on one road and when traffic flow is in aspecified range) is bound to be small. Also, low traffic flowsoccur mostly during the night, and can not be used to estimatethe safety performance function for the day. Also, peak hourdrivers are safer en-route to work than on their way home in theafternoon, and off-peak drivers tend to be a different lotaltogether.

7.1.4.2 Models

The first step of an empirical study of the relationship betweentraffic flow and accident frequency is to assemble, plot andexamine the data. The next step is to select the candidate modelequation(s) which might fit the data and serve as the safetyperformance function. Satterthwaite (1981, section 3) reviewsthe most commonly used models. Only those that are plausibleand depend on traffic flow only are listed below. Traffic flow,while important, is but one of the many variables on which theexpected accident frequency depends. However, since themonograph is devoted to traffic flow, the dependence on othervariables will not be pursued here. The Greek letters are theunknown parameters to be estimated from data.

��

� � �

When only one traffic stream is relevant the power function and The common feature of the models most often used is that theypolynomial models have been used: are linear or can be made so by logarithmization. This simplifies

m = �q (7.1)�

m = �q + �q +... (7.2)2

At times the more complex power form

m = �q (7.1a)�+�log(q)

is used which is also akin to the polynomial model 7.2 whenwritten in logarithms a straight line even for the higher flows where p(q) is not

constant any more. Similarly, if logic says that �=2, thelog(m) = log(�) + �log(q) + �[log(q)] (7.2a) quadratic growth applies for all q. In short, if � is selected to2

When two or more traffic streams or kinds of vehicles are from the origin. Conversely, if � is selected to fit the data best,relevant, the product of power functions seems common: the logical requirements will not be met.

m = �q q ... (7.3)1 2� �

statistical parameter estimation. The shapes of these functionsare shown in Figure 7.2.

The power function (Equation 7.1) is simple and can well satisfythe logical requirements near the origin (namely, that when q=0m=0, and that �=1 when one flow is involved or �=2 when twoflows are involved). However, its simplicity is also its downfall.If logic dictates, eg., that near the origin �=1 (say, for single-vehicle accidents), then the safety performance function has to be

meet requirements of logic, the model may not fit the data further

Figure 7.2Shapes of Selected Model Equations.

��

� � �

The popularity of the power function in empirical researchderives less from its suitability than from it being ‘lazy userfriendly’; most software for statistical parameter estimation canaccommodate the power function with little effort. Thepolynomial model (Equation 7.2) never genuinely satisfies thenear-origin requirements. Its advantage is that by using moreterms (and more parameters) the curve can be bent and shapedalmost at will. This is achieved at the expense of parsimony inparameters.

If the data suggest that as flow increases beyond a certain level,the slope of m(q) is diminishing, perhaps even grows negative,an additional model that is parsimonious in parameters mightdeserve consideration:

m=�q e (7.4)k �q

where k = 1 or 2 in accord with the near-origin requirements.When �<0 the function has a maximum at q = -k/�. Its formwhen k=1 and k=2 is shown in Figure 7.3. The advantage of thismodel is that it can meet the near-origin requirements and stillcan follow the shape of the data.

A word of caution is in order. In the present context the focus ison how accident frequency depends on traffic flow. Accordingly,the models were written with flow (q) as the principalindependent variable. However, traffic flow is but one of themany causal factors which affect accident frequency. Roadgeometry, time of day, vehicle fleet, norms of behaviour and thelike all play a part. Therefore, what is at times lumped into asingle parameter (� in equation 7.1, 7.1a, 7.3 and 7.4) reallyrepresents a complex multivariate expression. In short, themodeling of accident frequency is multivariate in nature.

Figure 7.3Two Forms of the Model in Equation 7.4.

��

� � �

7.1.4.3 Parameter Estimates in model 7.1 is 0.58; for rear-end accidents the exponent of

With the data in hand and the functional form selected, the nextstep is to estimate the parameters (�, �,...). In earlier work,estimation was often by minimization of squared deviations.This practice now seems deficient. Recognizing the discretenature of accident counts, the fact that their variance increaseswith their mean, and the possible existence of over-dispersion,it now seems that more appropriate statistical techniques arecalled for (see eg., Hauer 1992, Miaou and Lum 1993).

Results of past research are diverse. Part of the diversity is dueto the problems which beset both the cross-section and the timesequence of studies; another part is due to the use of AADT andsimilar long-period averages that have a less than direct tie toaccident occurrence; some of the diversity may come fromvarious methodological shortcomings (focus on accident rates,choice of simplistic model equations, use of inappropriatestatistical techniques); and much is due to the diversity betweenjurisdictions in what counts as a reportable accident and in theproportion of reportable accidents that get reported. Hauer andPersaud (1996) provide a comprehensive review of safetyperformance functions and their parameter values (for two-laneroads, multi-lane roads without access control, freeways,intersections and interchanges) based on North American data.A brief summary of this information and some internationalresults are given below.

A. Road Sections. In a cross-section study of Danish rural + (.94 × 10 )ADT for six-lane freeways and m = (5.4 ×roads Thorson (1967) estimated the exponent of ADT to be0.7. In a similar study of German rural roads Pfundt (1968)estimated the exponent of ADT to be 0.85. Kihlberg andTharp (1968) conducted an extensive cross-section studyusing data from several states. For sections that are 0.5 mileslong, they estimate the parameters for a series of road typesand geometric features. The model used is an elaboratedpower function m=�(ADT) (ADT) . The report� �log(ADT)

contains a rich set of results but creates little order in theotherwise bewildering variety. Ceder and Livneh (1982)used both cross-sectional and time-sequence data forinterurban road sections in Israel, using the simple powerfunction model (Equation. 1). The diverse results are difficultto summarize. Cleveland et al. (1985) divide low-volumerural two-lane roads into ‘bundles’ by geometry and find theADT exponents to range from 0.49 to 0.93 for off-roadaccidents Recent studies in the UK show that on urban roadsections, for single-vehicle accidents the exponent of AADT

AADT is 1.43. In a time-sequence study, Hall and Pendelton(1990) use ten mile 2 and 4-lane road segments surroundingpermanent counting stations in New Mexico and provide awealth of information about accident rates in relations tohourly flows and time of day. In an extensive cross-sectionstudy, Zegeer et al. (1986) find that the exponent of ADT is0.88 for the total number of accidents on rural two-laneroads. Ng and Hauer (1989) use the same data as Zegeerand show that the parameters differ from state to state andalso by lane width. For non-intersection accidents on ruraltwo lane roads in New York State, Hauer et al. (1994) foundthat when m is measured in [accidents/(mile-year)] andAADT is used for q, then in model 1, in 13 years � variesfrom 0.0024-0.0028 and �=0.78. Persaud (1992) using dataon rural roads in Ontario finds the exponent of AADT to varybetween 0.73 and 0.89, depending on lane and shoulderwidth. For urban two-lane roads in Ontario the exponent is0.72 For urban multi-lane roads (divided or undivided)�=1.14, for rural multi-lane divided roads it is 0.62 but forundivided roads it is again 1.13.

For California freeways Lundy (1965) shows that theaccidents per million vehicle miles increase roughly linearlywith ADT. This implies the quadratic relationship of model2. Based on the figures in Slatterly and Cleveland (1969),with m measured in [accidents/day], m = (5.8×10 )ADT +-7

(2.4×10 )ADT for four-lane freeways, m = (6.6×10 )ADT-11 2 -7

-11 2

10 ) × ADT + (.78 × 10 ) × ADT for eight-lane-7 -11 2

freeways. Leutzbach (1970) examines daytime accidents ona stretch of an autobahn. Fitting a power function to hisFigure (1c) and with m measured in accidents per day,m =(3×10 )×(hourly flow) . However, there is an indication-11 3

in this and other data that as flow increases, the accident rateinitially diminishes and then increases again. If so a thirddegree polynomial might be a better choice. Jovanis andChang (1987) fit model 7.3 to the Indiana Toll Road and findthe exponents to be 0.25 and 0.23 for car and truck-miles.Persaud and Dzbik (1993) find that when yearly accidentsare related to AADT, m=0.147×(AADT/1000) for 4-lane1.135

freeways but, when hourly flows are related toaccidents/hour, m = 0.00145 × (hourly flow/1000) . 0.717

Huang et al. (1992) report for California that Number ofaccidents = 0.65 + 0.666 × million-vehicle-miles.

��

� � �

B. Intersections. Tanner (1953) finds for rural T intersectionsin the UK the exponents to be 0.56 and 0.62 for left turning turning vehicles have to find gaps in the opposing traffic,traffic and main road traffic respectively. Roosmark (1966) m = 1.29 × 10 (hourly flow of left-turning cars) ×finds for similar intersections in Sweden the corresponding (hourly pedestrian flow) .exponents to be 0.42 and 0.71. For intersections on dividedhighways McDonald (1953) gives m in accidents per year asm = 0.000783 × (major road ADT) × (cross road0.455

ADT) . For signalized intersections in California, Webb0.633

(1955) gives m in accidents per year, as m = 0.00019 ×(major road ADT) × (cross road ADT) in urban areas0.55 0.55

with speeds below 40 km/h ; = 0.0056 × (major roadADT) × (cross road ADT) in semi-urban areas with0.45 0.38

speeds in the 40-70 km/h range; and = 0.007 × (majorroad ADT) × (cross road ADT) in rural areas with0.51 0.29

speeds above 70 km/h. For rural stop-controlled intersectionsin Minnesota, Bonneson and McCoy (1993) give m [inaccidents/year] = 0.692 × (major road ADT/1000) ×0.256

(cross road ADT/1000) . Recent studies done in the UK0.831

show that at signalized intersections for single vehicleaccidents the exponent of AADT in model 1 is 0.89; for rightangle accidents (model 7.3) the exponents of AADT are 0.36and 0.60 and for accident to left-turning traffic (their rightturn) the exponents are 0.57 and 0.46. Using data fromQuebec, Belangér finds that the expected annualnumber of accidents at unsignalized rural intersections is0.002×(Major road ADT) ×(Cross road ADT) .0.42 0.51

C. Pedestrians. Studies in the UK show that for nearsidepedestrian accidents on urban road sections the exponents forvehicle and pedestrian AADTs in model 7.3 are 0.73 and0.42. With m measured in [pedestrian accidents/year]Brüde and Larsson (1993) find that, at intersections,m = (7.3 × 10 )(incoming traffic/day) (crossing-6 0.50

pedestrians/day) . With m measured in (pedestrian0.7

accidents/hour), Quaye et al. (1993) find that if left-turningvehicles at signalized intersections do not face opposingvehicular traffic then m = 1.82 × 10 (hourly flow of left--8

turning cars) × (hourly pedestrian flow) ; when the left1.32 0.34

-7 0.36

0.86

7.1.5 Closure

Many aspects of the traffic stream are related to the frequencyand severity of accidents; only the relationship with flow hasbeen discussed here. How safety depends on flow is importantto know. The relationship of traffic flow to accident frequencyis called the ‘safety performance function’. Only when the safetyperformance function is known, can one judge whether one kindof design is safer than another, or whether an intervention hasaffected the safety of a facility. Simple division by flow tocompute accident rates is insufficient because the typical safetyperformance function is not linear.

Past research about safety performance functions has led todiverse results. This is partly due to the use of flow data whichare an average over a long time period (such as AADT), partlydue to the difficulties which are inherent in the cross-sectionaland the time-sequence studies, and partly because accidentreporting and roadway definitions vary among jurisdictions.However, a large part of the diversity is due to the fact thataccident frequency depends on many factors in addition to trafficflow and that the dependence is complex.Today, some of the pastdifficulties can be overcome. Better information about trafficflows is now available (eg. from freeway-traffic-management-systems, permanent counting stations, or weight-in-motiondevices); also better methods for the multivariate analysis ofaccident counts now exist. However, in addition to progress instatistical modeling, significant advances seem possible throughthe infusion of detailed theoretical modeling which makes use ofall relevant characteristics of the traffic stream such as speed,flow, density, headways and shock waves.

7.2 Fuel Consumption Models

Substantial energy savings can be achieved through urban traffic saved each day. It is further estimated that 45 percent of the totalmanagement strategies aimed at improving mobility and energy consumption in the U.S. is by vehicles on roads. Thisreducing delay. It is conservatively estimated, for example, that amounts to some 240 million liters (63 million gallons) ofif all the nearly 250,000 traffic signals in the U.S. were optimally petroleum per day, of which nearly one-half is used by vehiclestimed, over 19 million liters (5 million gallons) of fuel would be under urban driving conditions.

��

� � �

Fuel consumption and emissions have thus become increasingly pavement roughness, grades, wind, altitude, etc. is hoped to beimportant measures of effectiveness in evaluating traffic small due to collective effects if data points represent aggregatemanagement strategies. Substantial research on vehicular energy values over sufficiently long observation periods. Alternatively,consumption has been conducted since the 1970's, resulting in model estimates may be adjusted for known effects of roadwayan array of fuel consumption models. In this section, a number grade, ambient temperature, altitude, wind conditions, payload,of such models which have been widely adopted are reviewed. etc.

7.2.1 Factors Influencing VehicularFuel Consumption

Many factors affect the rate of fuel consumption. These factorscan be broadly categorized into four groups: vehicle,environment, driver, and traffic conditions. The main variablesin the traffic category include speed, number of stops, speednoise and acceleration noise. Speed noise and acceleration noisemeasure the amount of variability in speed and acceleration interms of the variance of these variables. The degree of driveraggressiveness also manifests itself in speed and accelerationrates and influences the fuel consumption rate.

Factors related to the driving environment which could affectfuel consumption include roadway gradient, wind conditions,ambient temperature, altitude, and pavement type (for example,AC/PCC/gravel) and surface conditions (roughness, wet/dry).Vehicle characteristics influencing energy consumption includetotal vehicle mass, engine size, engine type (for example,gasoline, diesel, electric, CNG), transmission type, tire type andsize, tire pressure, wheel alignment, the status of brake andcarburetion systems, engine temperature, oil viscosity, gasolinetype (regular, unleaded, etc.), vehicle shape, and the degree ofuse of auxiliary electric devices such as air-conditioning, radio,wipers, etc. A discussion of the degree of influence of most ofthe above variables on vehicle fuel efficiency is documented bythe Ontario Ministry of Transportation and Communications(TEMP 1982).

7.2.2 Model Specifications

Fuel consumption models are generally used to evaluate theeffectiveness and impact of various traffic managementstrategies. As such these models are developed using datacollected under a given set of vehicle fleet and performancecharacteristics such as weight, engine size, transmission type,tire size and pressure, engine tune-up and temperatureconditions, etc. The variation in fuel consumption due to otherfactors such as driver characteristics, ambient temperature,

Given a fixed set of vehicle and driver characteristics andenvironmental conditions, the influence of traffic-related factorson fuel consumption can be modeled. A number of studies inGreat Britain (Everall 1968), Australia (Pelensky et al. 1968),and the United States (Chang et al. 1976; Evans and Herman1978; Evans et al. 1976) all indicated that the fuel consumptionper unit distance in urban driving can be approximated by alinear function of the reciprocal of the average speed. One suchmodel was proposed by Evans, Herman, and Lam (1976), whostudied the effect of sixteen traffic variables on fuel consumption.They concluded that speed alone accounts for over 70 percent ofthe variability in fuel consumption for a given vehicle.Furthermore, they showed that at speeds greater than about 55km/h, fuel consumption rate is progressively influenced by theaerodynamic conditions. They classified traffic conditions asurban (V < 55 km/h) versus highway (V > 55 km/h) trafficshowing that unlike the highway regime, in urban driving fuelefficiency improves with higher average speeds (Figure 7.4).

7.2.3 Urban Fuel Consumption Models

Based on the aforementioned observations, Herman and co-workers (Chang and Herman 1981; Chang et al. 1976; Evansand Herman 1978; Evans et al. 1976) proposed a simpletheoretically-based model expressing fuel consumption in urbanconditions as a linear function of the average trip time per unitdistance (reciprocal of average speed). This model, known asthe Elemental Model, is expressed as:

� = K + K T, V < 55 km/hr (7.5)1 2where,

� : fuel consumption per unit distance T : average travel time per unit distance

andV(=1/T) : average speed

K and K are the model parameters. K (in mL/km) represents1 2 1fuel used to overcome the rolling friction and is closely related

��

� � ��

Figure 7.4Fuel Consumption Data for a Ford Fairmont (6-Cyl.)

Data Points represent both City and Highway Conditions.

to the vehicle mass (Figure 7.5). K (in mL/sec) is a function In F = average fuel consumption per roadway section2an effort to improve the accuracy of the Elemental Model, other (mL)researchers have considered additional independent variables.Among them, Akcelik and co-workers (Akcelik 1981;Richardson and Akcelik1983) proposed a model whichseparately estimates the fuel consumed in each of the threeportions of an urban driving cycle, namely, during cruising,idling, and deceleration-acceleration cycle. Hence, the fuelconsumed along an urban roadway section is estimated as:

F = f X + f d + f h, (7.11)1 s 2 s 3

where,

X = total section distance (km)s

d = average stopped delay per vehicle (secs)s

h = average number of stops per vehiclef = fuel consumption rate while cruising (mL/km)1

f = fuel consumption rate while idling (mL/sec)2

f = excess fuel consumption per vehicle stop (mL)3

The model is similar to that used in the TRANSYT-7Fsimulation package (Wallace 1984).

Herman and Ardekani (1985), through extensive field studies,have shown that delay and number of stops should not be used

��

� � ��

Figure 7.5Fuel Consumption

Versus Trip Time per Unit Distance for a Number of Passenger Car Models.

together in the same model as estimator variables. This is due where,to the tendency for the number of stops per unit distance to behighly correlated with delay per unit distance under urban trafficconditions. They propose an extension of the elemental modelof Equation 7.5 in which a correction is applied to the fuelconsumption estimate based on the elemental model dependingon whether the number of stops made is more or less than theexpected number of stops for a given average speed (Hermanand Ardekani 1985).

A yet more elaborate urban fuel consumption model has been setforth by Watson et al. (1980). The model incorporates thechanges in the positive kinetic energy during acceleration as apredictor variable, namely,

F = K + K /V + K V + K PKE, (7.12)1 2 s 3 s 4

where,F = fuel consumed (Lit/km)V = space mean speed (km/hr)s

The term PKE represents the sum of the positive kinetic energychanges during acceleration in m/sec , and is calculated as2

follows:

PKE =�(V - V )/(12,960 X ) (7.13)f i s2 2

V = final speed (km/hr)f

V = initial speed (km/hr)i

X = total section length (km)s

A number of other urban fuel consumption models are discussedby other researchers, among which the work by Hooker et al.(1983), Fisk (1989), Pitt et al. (1987), and Biggs and Akcelik(1986) should be mentioned.

7.2.4 Highway Models

Highway driving corresponds to driving conditions under whichaverage speeds are high enough so that the aero-dynamic effectson fuel consumption become significant. This occurs at averagespeeds over about 55 km/h (Evans et al. 1976). Two highwaymodels based on constant cruising speed are those by Vincent etal. (1980) and Post et al. (1981). The two models are valid atany speed range, so long as a relatively constant cruise speed canbe maintained (steady state speeds). The steady-state speedrequirement is, of course, more easily achievable under highwaydriving conditions.

0 75 150 225 300 375

120

240

360

480

Average Trip Time Unit Distance, T (Secs/ km)

1974 Standard -Sized car

1976 Subcompact car

��

� � ��

Figure 7.6Fuel Consumption Data and the Elemental Model Fit

for Two Types of Passenger Cars (Evans and Herman 1978).

The model by Vincent, Mitchell, and Robertson is used in the Calibration of this model for a Melbourne University test carTRANSYT-8 computer package (Vincent et al. 1980) and is in (Ford Cortina Wagon, 6-Cyl, 4.1L, automatic transmission)the form: yields (Akcelik 1983) the following parameter values (Also see

f = a + b V + c V , c c c2

(7.14) b = 15.9 mL/km

where,

V = steady-state cruising speed (km/hr)c

f = steady-state fuel consumption rate at cruising estimate fuel consumption for non-steady-state speed conditionscspeed (mL/km), calibration of this model for a under both urban or highway traffic regimes. These models aremid-size passenger car yields used in a number of microscopic traffic simulation packages

a = 170 mL/km, b = -4.55 mL-hr/km ; and2

c = 0.049 mL-hr /km (Akcelik 1983).2 3

A second steady-state fuel model formulated by Post et al.(1981) adds a V term to the elemental model of Equation 7.5 to2

account for the aero-dynamic effects, namely,

f = b + b /V + b V . (7.15)c 1 2 c 3 c2

Figure 7.7):

1

b = 2,520 mL/hr2

b = 0.00792 mL-hr /km .32 3

Instantaneous fuel consumption models may also be used to

such as NETSIM (Lieberman et al. 1979) and MULTSIM(Gipps and Wilson 1980) to estimate fuel consumption based oninstantaneous speeds and accelerations of individual vehicles.

By examining a comprehensive form of the instantaneous model,Akcelik and Bayley (1983) find the following simpler form ofthe function to be adequate, namely,

f = K + K V + K V + �K aV + K a V � (7.16)1 2 3 4 53 2

0 20 40 60 80 100

50

100

150

200

250

300

Constant Cruise Speed, Vc (km/h)

��

� � ��

Figure 7.7Constant-Speed Fuel Consumption per Unit Distance for the Melbourne University Test Car (Akcelik 1983).

where,f = instantaneous fuel consumption (mL/sec)V = instantaneous speed (km/hr)a = instantaneous acceleration rate

(a > 0)(km/hr/sec)K = parameter representing idle flow rate1

(mL/sec)K = parameter representing fuel consumption2

to overcome rolling resistanceK = parameter representing fuel consumption3

to overcome air resistance K K = parameters related to fuel consumption due4, 5

to positive acceleration

The above model has been used by Kenworthy et al. (1986) toassess the impact of speed limits on fuel consumption.

7.2.5 Discussion

During the past two decades, the U.S. national concerns overdependence on foreign oil and air quality have renewedinterest in vehicle fuel efficiency and use of alternative fuels.Automotive engineers have made major advances in vehicularfuel efficiency (Hickman and Waters 1991; Greene and Duleep1993; Komor et al. 1993; Greene and Liu 1988).

Such major changes do not however invalidate the modelspresented, since the underlying physical laws of energyconsumption remain unchanged. These include the relationbetween energy consumption rate and the vehicle mass, enginesize, speed, and speed noise. What does change is the need torecalibrate the model parameters for the newer mix of vehicles.This argument is equally applicable to alternative fuel vehicles(DeLuchi et al. 1989), with the exception thatthere may also bea need to redefine the variable units, for example, from mL/secor Lit/km to KwH/sec or KwH/km, respectively.

��

� � ��

7.3 Air Quality Models

7.3.1 Introduction

Transportation affects the quality of our daily lives. It influencesour economic conditions, safety, accessibility, and capability toreach people and places. Efficient and safe transportationsatisfies us all but in contrast, the inefficient and safe use of ourtransportation system and facilities which result in trafficcongestions and polluted air produces personal frustration andgreat economic loss.

The hazardous air pollutants come from both mobile andstationary sources. Mobile sources include passenger cars, lightand heavy trucks, buses, motorcycles, boats, and aircraft.Stationary sources range from oil refineries to dry cleaners andiron and steel plants to gas stations.

This section concentrates on mobile source air pollutants whichcomprise more than half of the U.S. air quality problems.Transportation and tailpipe control measure programs inaddition to highway air quality models are discussed in thesection.

Under the 1970 U.S. Clean Air Act, each state must prepare aState Implementation Plan (SIP) describing how it will controllemissions from mobile and stationary sources to meet therequirements of the National Ambient Air Quality Standards(NAAQS) for six pollutants: (1) particulate matter (formerlyknown as total suspended particulate (TSP) and now as PM10which emphasizes the smaller particles), (2) sulfur dioxide(SO2), (3) carbon monoxide (CO), (4) nitrogen dioxide (NO2),(5) ozone (O3), and (6) lead (Pb).

The 1990 U.S. Clean Air Act requires tighter pollution standardsespecially for cars and trucks and would empower theEnvironmental Protection Agency (EPA) to withhold highwayfunds from states which fail to meet the standards for major airpollutants (USDOT 1990). The 1970 and 1990 federal emissionstandards for motor vehicles are shown in Table 7.1.

This section concentrates on mobile source air pollutants whichcomprise more than half of the U.S. air quality problems.Exhaust from these sources contain carbon monoxide, volatileorganic compounds (VOCs), nitrogen oxides, particulates, andlead. The VOCs along with nitrogen oxides are the majorelements contributing to the formation of "smog".

Carbon monoxide which is one of the main pollutants is acolorless, and poisonous gas which is produced by theincomplete burning of carbon in fuels. The NAAQS standardfor ambient CO specifies upper limits for both one-hour andeight-hour averages that should not be exceeded more than onceper year. The level for one-hour standard is 35 parts per million(ppm), and for the eight-hour standard is 9 ppm. Mostinformation and trends focus on the 8-hour average because it isthe more restrictive limit (EPA 1990).

7.3.2 Air Quality Impacts of TransportationControl Measures

Some of the measures available for reducing traffic congestionand improving mobility and air quality is documented in a reportprepared by Institute of Transportation Engineers (ITE) in 1989.This "toolbox" cites, as a primary cause of traffic congestion, theincreasing number of individuals commuting by automobile inmetropolitan areas, to and from locations dispersed throughouta wide region, and through areas where adequate highwaycapacity does not exist. The specific actions that can be taken toimprove the situation are categorized under five components asfollows:

1) Getting the most out of the existing highway system- Intelligent Transportation Systems (ITS)- urban freeways (ramp metering, HOV's)- arterial and local streets (super streets, parking

management)- enforcement

2) Building new capacity (new highway, reconstruction)3) Providing transit service (paratransit service, encouraging

transit use)4) Managing transportation demand

- strategic approaches to avoiding congestion (roadpricing)

- mitigating existing congestion (ridesharing)5) Funding and institutional measures

- funding (fuel taxes, toll roads)- institutional measures (transportation management

associations)

��

� � ��

Table 7.1Federal Emission Standards

1970 Standards 1990 Standards(grams/km) (grams/km)

Light Duty Vehicles (0-3, 340 Kgs)1

Carbon Monoxide (CO) 2.11 2.11

Hydrocarbons (HC) 0.25 0.16

Oxides of Nitrogen (NOx) 0.62 0.252

Particulates 0.12 0.05

Light Duty Vehicles (1,700-2,600 Kgs)1





1970 Standards 1990 Standards(grams/km) (grams/km)

Light Duty Trucks (over 2,600 Kgs GVWR)





Light duty vehicles include light duty trucks.1

The new emission standards specified in this table are for useful life of 5 years or 80,000 Kms whichever first occurs.2

7.3.3 Tailpipe Control Measures

It is clear that in order to achieve the air quality standards and toreduce the pollution from the motor vehicle emissions,substantial additional emission reduction measures are essential.According to the U.S. Office of Technology Assessment (OTA),the mobile source emissions are even higher in most sources -responsible for 48 percent of VOCs in non-attainment U.S. citiesin 1985, compared to other individual source categories (Walsh1989).

The following set of emission control measures, if implemented,has the potential to substantially decrease exhaust emissionsfrom major air pollutants:

� limiting gasoline volatility to 62.0 KPa (9.0 psi) RVP;� adopting "onboard" refueling emissions controls;� "enhanced" inspection and maintenance programs;

requiring "onboard diagnostics" for emission controlsystems;

� adopting full useful life (160,000 kms) requirements;

��

� � ��

� requiring alternative fuel usage; - humidity� utilizing oxygenated fuels/reformulated gasoline - wind and temperature fluctuations

measures; � traffic data� adopting California standards (Tier I and II). - traffic volume

Some of the measures recommended in California standards - vehicle length or typeinclude: improved inspection and maintenance such as � site types"centralized" inspection and maintenance programs, heavy duty - at-grade sitesvehicle smoke enforcement, establishing new diesel fuel quality - elevated sitesstandards, new methanol-fueled buses, urban bus system - cut siteselectrification, and use of radial tires on light duty vehicles. � period of measurement

In the case of diesel-fueled LDT's (0-3,750 lvw) and light-duty Some of these models which estimate the pollutant emissionsvehicles, before the model year 2004, the applicable standards from highway vehicles are discussed in more detail in thefor NOx shall be 0.62 grams/km for a useful life as defined following sections.above.

7.3.4 Highway Air Quality Models

As discussed earlier, federal, state, and local environmentalregulations require that the air quality impacts of transportation -related projects be analyzed and be quantified. For this purpose,the Federal Highway Administration (FHWA) has issuedguidelines to ensure that air quality effects are considered duringplanning, design, and construction of highway improvements, sothat these plans are consistent with State Implementation Plans(SIPs) for achieving and maintaining air quality standards .

For example, the level of CO associated with a given project isa highway - related air quality impact that requires evaluation.In general, it must be determined whether the ambient standardsfor CO (35 ppm for 1 hour and 9 ppm for 8 hours, not to beexceeded more than once per year) will be satisfied or exceededas a result of highway improvements. This requirement calls forestimating CO concentrations on both local and areawide scales.

A number of methods of varying sophistication and complexityare used to estimate air pollutant levels. These techniquesinclude simple line-source-oriented Gaussian models as well asmore elaborate numerical models (TRB 1981). The databasesused in most models could be divided into the followingcategories:

� meteorological data- wind speed and direction- temperature

- vehicle speed

7.3.4.1 UMTA Model

The most simple of these air quality models is the one whichrelates vehicular speeds and emission levels (USDOT 1985).The procedure is not elaborate but is a quick-response techniquefor comparison purposes. This UMTA (now Federal TransitAdministration) model contains vehicular emission factorsrelated to speed of travel for freeways and surfaced arterials.

The model uses a combination of free flow and restrained (peakperiod) speeds. It assumes that one-third of daily travel wouldoccur in peak hours of flow reflecting restrained (congested)speeds, while two-thirds would reflect free-flow speedcharacteristics. For a complete table of composite emissionfactors categorized by autos and trucks for two calendar years of1987 and 1995 refer to Characteristics of Urban TransportationSystem, U.S. Department of Transportation, Urban MassTransportation Administration, October 1985.

7.3.4.2 CALINE-4 Dispersion Model

This line source air quality model has been developed by theCalifornia Department of Transportation (FHWA 1984). It isbased on the Gaussian diffusion equation and employs a mixingzone concept to characterize pollutant dispersion over theroadway.

The model assesses air quality impacts near transportationfacilities given source strength, meteorology, and site geometry.CALINE-4 can predict pollution concentrations for receptors

��

� � ��

located within 500 meters of the roadway. It also has special � distance from link endpoints to stopline;options for modeling air quality near intersections, street � acceleration and deceleration times (ACCT, DCLT);canyons, and parking facilities. � idle times at front and end of queue;

CALINE-4 uses a composite vehicle emission factor in grams � idle emission rate (EFI).per vehicle-mile and converts it to a modal emission factor. TheEnvironmental Protection Agency (EPA) has developed a seriesof computer programs, the latest of which is called Mobile4.1(EPA 1991), for estimating composite mobile emission factorsgiven average route speed, percent cold and hot-starts, ambienttemperature, vehicle mix, and prediction year. These emissionfactors are based on vehicle distribution weighted by type, age,and operation mode, and were developed from certification andsurveillance data, mandated future emission standards, andspecial emission studies.

Composite emission factors represent the average emission rateover a driving cycle. The cycle might include acceleration,deceleration, cruise, and idle modes of operation. Emission ratesspecific to each of these modes are called modal emissionfactors. The speed correction factors used in composite emissionfactor models, such as MOBILE4, are derived from variabledriving cycles representative of typical urban trips. The FederalTest Procedures (FTP) for driving cycle are the basis for most ofthese data.

Typical input variables for the CALINE-4 model are shown inTable 7.2. In case of an intersection, the following assumptionsare made for determining emission factors:

� uniform vehicle arrival rate;� constant acceleration and deceleration rates; constant time

rate of emissions over duration of each mode;� deceleration time rate of emissions equals 1.5 times the idle

rate;� an "at rest" vehicle spacing of 7 meters; and� all delayed vehicles come to a full stop.

In addition to composite emission factor at 26 km/hr (EFL), thefollowing variables must be quantified for each intersection link:

� arrival volume in vehicles per hour;� departure volume in vehicles per hour;� average number of vehicles per cycle per lane for the

dominant; movement � average number or vehicles delayed per cycle per lane for

the dominant movement (NDLA);

� cruise speed (SPD); and

The following computed variables are determined for each linkfrom the input variables:

� acceleration rate;� deceleration rate;� acceleration length; � deceleration length; � acceleration-speed product;� FTP-75 (BAG2) time rate emission factor;� acceleration emission factor; � cruise emission factor; � deceleration emission factor; and� queue length (LQU=NDLA*VSP), where VSP is the "at

rest" vehicle spacing.

The cumulative emission profiles (CEP) for acceleration,deceleration, cruise, and idle modes form the basis fordistributing the emissions. These profiles are constructed foreach intersection link, and represent the cumulative emissionsper cycle per lane for the dominant movement. The CEP isdeveloped by determining the time in mode for each vehicleduring an average cycle/lane event multiplied by the modalemission time rate and summed over the number of vehicles.CALINE4 can predict concentrations of relatively inertpollutants such as carbon monoxide (CO), and other pollutantslike nitrogen dioxide (NO2) and suspended particles.

7.3.4.3 Mobie Source Emission Factor Model

MOBILE4.1 is the latest version of mobile source emissionfactor model developed by the U.S. Environmental ProtectionAgency (EPA). It is a computer program that estimateshydrocarbon (HC), carbon monoxide (CO), and oxides ofnitrogen (NOx) emission factors for gasoline-fueled and diesel-fueled highway motor vehicles.

MOBILE4.1 calculates emission factors for eight vehicle typesin two regions (low- and high-altitude). Its emission estimatesdepend on various conditions such as ambient temperature,speed, and mileage accrual rates. MOBILE4.1 will estimateemission factors for any calendar year between 1960 and 2020.

��

� � ��

Table 7.2Standard Input Values for the CALINE4

I. Site Variables

Temperature ©

Wind Speed (m/s)

Wind Direction (deg)

Directional Variability (deg)

Atmospheric Stability (F)

Mixing Height (m)

Surface Roughness (cm)

Settling Velocity (m/s)

Deposition Velocity (m/s)

Ambient Temperature ©

II. Link Variables

Traffic Volume (veh/hr)

Emission Factor (grams/veh-mile)

Height (m)

Width (m)

Link Coordinates (m)

III. Receptor Locations (m)

The 25 most recent model years are considered to be in Speed correction factors are used by the model to correct exhaustoperation in each calendar year. It is to be used by the states in emissions for average speeds other than that of the FTP (32the preparation of the highway mobile source portion of the 1990 km/hr). MOBILE4.1 uses three speed correction models: lowbase year emission inventories required by the Clean Air Act speeds (4-32 km/hr), moderate speeds (32-77 km/hr), and highAmendments of 1990. speeds (77-105 km/hr). The pattern of emissions as a function

MOBILE4.1 calculates emission factors for gasoline-fueled model year groups. Emissions are greatest at the minimumlight-duty vehicles (LDVs), light-duty trucks (LDTs), heavy-duty speed of 4 km/hr, decline relatively rapidly as speeds increasevehicles (HDVs), and motorcycles, and for diesel LDVs, LDTs, from 4 to 32 km/hr, decline more slowly as speeds increase fromand HDVs. It also includes provisions for modeling the effects 32 to 77 km/hr, and then increase with increasing speed to theof oxygenated fuels (gasoline-alcohol and gasoline- ether blends) maximum speed of 105 km/hr.on exhaust CO emissions. Some of the primary input variablesand their ranges are discussed below.

of vehicle speed is similar for all pollutants, technologies, and

��

� � ��

The vehicle miles traveled (VMT) mix is used to specify the based on interpolation of the calendar year 1990 and 1991fraction of total highway VMT that is accumulated by each of the MOBILE4.1 emission factors. eight regulated vehicle types. The VMT mix is used only tocalculate the composite emission factor for a given scenario on One important determinant of emissions performance is thethe basis of the eight vehicle class-specific emission factors. mode of operation. The EPA's emission factors are based onConsidering the dependence of the calculated VMT mix on the testing over the FTP cycle, which is divided into three segmentsannual mileage accumulation rates and registration distributions or operating modes: cold start, stabilized, and hot start.by age, EPA expects that states develop their own estimates of Emissions generally are highest when a vehicle is in the cold-VMT by vehicle type for specific highway facility, sub-zones, start mode: the vehicle, engine, and emission control equipmenttime of day, and so on. are all at ambient temperature and thus not performing at

Many areas of the country have implemented inspection and start mode, when the vehicle is not yet completely warmed upmaintenance (I/M) programs as a means of further reducing but has not been sitting idle for sufficient time to have cooledmobile source air pollution. MOBILE4.1 has the capability of completely to ambient temperature. Finally, emissions generallymodeling the impact of an operating I/M program on the are lowest when the vehicle is operating in stabilized mode, andcalculated emission factors, based on user specification of has been in continuous operation long enough for all systems tocertain parameters describing the program to be modeled. Some have attained relatively stable, fully "warmed-up" operatingof the parameters include: temperatures.

� program start year and stringency level; The EPA has determined through its running loss emission test� first and last model years of vehicles subject to programs that the level of running loss emissions depends on

program; several variables: the average speed of the travel, the ambient� program type (centralized or decentralized); temperature, the volatility (RVP) of the fuel, and the length of� frequency of inspection (annual or biennial); and the trip. "Trip length" as used in MOBILE4.1 refers to the� test types. duration of the trip (how long the vehicle has been traveling), not

MOBILE4.1 (EPA 1991) has the ability to model uncontrolled driven). Test data show that for any given set of conditionslevels of refueling emissions as well as the impacts of the (average speed, ambient temperature, and fuel volatility),implementation of either or both of the major types of vehicle running loss emissions are zero to negligible at first, but increaserecovery systems. These include the "Stage II" (at the pump) significantly as the duration of the trip is extended and the fuelcontrol of vehicle refueling emissions or the "onboard" (on the tank, fuel lines, and engine become heated.vehicle) vapor recovery systems (VRS).

The minimum and maximum daily temperatures are used in 7.3.4.4 MICRO2MOBILE4.1 in the calculation of the diurnal portion ofevaporative HC emissions, and in estimating the temperature ofdispensed fuel for use in the calculation of refueling emissions.The minimum temperature must be between -18 C to 38 C (0 Fand 100 F), and the maximum temperature must be between -12C to 49 C (10 F and 120 F) inclusive.

The value used for calendar year in MOBILE4.1 defines the yearfor which emission factors are to be calculated. The model hasthe ability to model emission factors for the year 1960 through2020 inclusive. The base year (1990) inventories are based ona typical day in the pollutant season, most commonly Summerfor ozone and Winter for CO. The base year HC inventories are

optimum levels. Emissions are generally somewhat lower in hot

on the distance traveled in the trip (how far the vehicle has been

MICRO2 is an air quality model which computes the airpollution emissions near an intersection. The concentration ofthe pollutants in the air around the intersection is not computed.In order to determine the pollution concentration, a dispersionmodel which takes weather conditions such as wind, speed, anddirection into account should be used (Richards 1983).

MICRO2 bases its emissions on typical values of the FTPperformed for Denver, Colorado in the early 1980s. They are:

� FTP (1) HC 6.2 grams/veh/km� FTP (2) CO 62.2 grams/veh/km

��

� � ��

� FTP (3) NOx 1.2 grams/veh/km

For lower than Denver altitudes or years beyond 1980's,emission rates may be lower and should change from these initialvalues.

The emission formulas as a function of acceleration and speedsare as follow:

HC Emission (gram/sec) =0.018 + 5.668*10 (A*S) + 2.165*10 (A*S ) (7.17a)-3 -4 2

CO Emission (gram/sec) =0.182 - 8.587*10 (A*S) + 1.279*10 (A*S ) (7.17b)-2 -2 2

NOx Emission (gram/sec) =3.86*10 + 8.767*10 (A*S) (for A*S>0) (7.17c)-3 -3

NOx Emission (gram/sec) =1.43*10 - 1.830*10 (A*S) (for A*S<0) (7.17d)-3 -4

where,A = acceleration (meters/sec ) and2

S = speed (meters/sec).

7.3.4.5 The TRRL Model

This model has been developed by the British Transport andRoad Research Laboratory (TRRL) and it predicts air pollutionfrom road traffic (Hickman and Waterfield 1984). Theestimations of air pollution are in the form of hourly averageconcentrations of carbon monoxide at selected locations arounda network of roads. The input data required are theconfiguration of the road network, the location of the receptor,traffic volumes and speeds, wind speed, and wind direction.

The concentration of carbon monoxide may be used as toapproximate the likely levels of other pollutants using thefollowing relations:

HC (ppm) = 1.8 CO (ppm) * R + 4.0 (7.18a)

NOx (ppm) = CO (ppm) * R + 0.1 (7.18b)

where R is the ratio of pollutant emission rate to that of carbonmonoxide for a given mean vehicle speed,

Mean Speed NOx HC(km/hr) (ppm) (ppm)

20 0.035 0.20530 0.050 0.24040 0.070 0.26050 0.085 0.28060 0.105 0.29070 0.120 0.305

A quick but less accurate estimate of the annual maximum 8hour CO concentration can also be obtained from the averagepeak hour CO concentration estimate, as follows:

C = 1.85 C + 1.19 (7.19)8 1

where C is the annual maximum 8 hour concentration, and C8 1is the average peak hour concentration.

A graphical screening test is introduced by which any propertieslikely to experience an air pollution problem are identified. Theprocedure first reduces the network to a system of long roads androundabouts (if any). Then from a graph, the concentration ofcarbon monoxide for standard traffic conditions for locations atany distance from each network element may be determined.Factors are then applied to adjust for the traffic conditions at thesite and the sum of the contributions from each element gives anestimate of the likely average peak hour concentration. Anexample of the graphical screening test results is shown in Table7.3.

7.3.5 Other Mobile SourceAir Quality Models

There are many other mobile source models which estimate thepollutant emission rates and concentrations near highway andarterial streets. Most of these models relate vehicle speeds andother variables such as vehicle year model, ambient temperature,and traffic conditions to emission rates. A common example ofthis type of relation could be found in Technical Advisory#T6640.10 of EPA report, "Mobile Source Emission FactorTables for MOBILE3." Other popular models include HIWAY2and CAL3QHC. HIWAY2 model has been developed by U.S.EPA to estimate hourly concentrations of non-reactivepollutants, like CO, downwind of roadways. It is usually used

��

� � ��

Table 7.3Graphical Screening Test Results for Existing Network

Distance from Centerline (m) 13 45 53 103

CO for 1000 vehs/hr at 100 km/h (ppm) 1.08 0.49 0.40 0.11

Traffic Flow (vehs/hr) 2,600 1,400 800 400

Speed (km/hr) 40 20 20 50

Speed Correction Factor 2.07 3.59 3.59 1.73

CO for Actual Traffic Conditions (ppm) 5.81 2.46 1.15 0.08

Total 1-Hour CO = 9.50(ppm) Equivalent Annual Maximum 8-Hour = 18.76(ppm)

for analyzing at- grade highways and arterials in uniform wind is especially designed to handle near-saturated and/or over-conditions at level terrain as well as at depressed sections (cuts) capacity traffic conditions and complex intersections whereof roadways. The model cannot be used if large obstructions major roadways interact through ramps and elevated highways.such as buildings or large trees hinder the flow of air. The The model combines the CALINE3 line source dispersion modelsimple terrain requirement makes this model less accurate for with an algorithm that internally estimates the length of theurban conditions than CALINE4 type of models. queues formed by idling vehicles at signalized intersections. The

The CAL3QHC model has the ability to account for the required by transportation models such as roadway geometries,emissions generated by vehicles traveling near roadway receptor locations, vehicular emissions, and meteorologicalintersections. Because idling emissions account for a substantial conditions. Emission factors used in the model should beportion of the total emissions at an intersection, this capability obtained from mobile source emission factor models such asrepresents a significant improvement in the prediction of MOBILE4.pollutant concentrations over previous models. This EPA model

inputs to the model includeinformation and data commonly

References

Akcelik, R. (1981). Fuel Efficiency and Other Objectives in Andreassen, D. (1991). Population and Registered VehicleTraffic System Management. Traffic Engineering & Data Vs. Road Deaths. Accident Analysis & Prev., Vol. 23,Control, Vol. 22, pp. 54-65. No. 5, pp. 343-351.

Akcelik, R. And C. Bayley (1983). Some Results on Fuel Belmont, D. M. (1953). Effect of Average Speed And VolumeConsumption Models. Appeared in Progress in Fuel On Motor-vehicle Accidents On Two-lane Tangents. Proc.Consumption Modelling for Urban Traffic Management.Research Report ARR No. 124, Austrailian Road ResearchBoard, pp. 51-56.

Highway Research Board, 32, pp. 383-395.

��

� � ��

Biggs, D. C. and R. Akcelik (1986). An Energy-Related Federal Highway Administration (1984). CALINE4 - AModel for Instantaneous Fuel Consumption. Traffic Dispersion Model for Predicting Air PollutantEngineering & Control, Vol.27, No.6, pp. 320-325.

Bonneson, J. A., and P. T. McCoy (1993). Estimation 84/15.of Safety At Two-way Stop-controlled Intersections On Fisk, C. S. (1989). The Australian Road Research BoardRural Highways. Transportation Research Record 1401, pp. Instantaneous Model of Fuel Consumption. Transportation83-89, Washington DC. Research, Vol. 23B, No. 5, pp. 373-385.

Brundell-Freij, K. and L. Ekman (1991). Flow and Safety. Gipps, P. G. and B. G. Wilson (1980). MULTISIM: APresented at the 70th Annual Meeting of the TransportationResearch Board, Washington, DC.

Brüde, U. and J. Larsson (1993). Models For PredictingAccidents At Junctions Where Pedestrians And Cyclists AreInvolved. How Well Do They Fit? Accident Analysis &Prev. Vol. 25, No. 6, pp. 499-509.

Ceder, A. and M. Livneh (1978). Further Evaluation Of TheRelationships Between Road Accidents and Average DailyTraffic. Accident Analysis & Prev., Vol 10, pp. 95-109.

Ceder, A. (1982). Relationship Between Road Accidents AndHourly Traffic Flow-II. Accident Analysis & Prev., Accident Analysis and Prevention, Vol. 3, No. 1, pp. 1-13.Vol 14, No. 1, pp. 35-44.

Ceder, A. and M. Livneh (1982). Relationship BetweenRoad Accidents And Hourly Traffic Flow-I. AccidentAnalysis & Prev., Vol. 14, No. 1.

Chang, M. F., L. Evans, R. Herman, and P. Wasielewski(1976). Gasoline Consumption in Urban Traffic.Transportation Research Record 599, TransportationResearch Board, pp. 25-30.

Chang, M. F. and R. Herman (1981). Trip Time Versus StopTime and Fuel Consumption Characteristics in Cities.Transportation Science, Vol. 15, No. 3, pp. 183-209.

Cleveland, D. E., L. P. Kostyniuk, and K.-L. Ting (1985). Design and Safety On Moderate Volume Two-lane Roads.Transportation Research Record, 1026, pp. 51-61.

DeLuchi, M., Q. Wang, and D. Sperling (1989). ElectricVehicles: Performance, Life-Cycle Costs, Emissions, andRecharging Requirements. Transportation Research,Vol. 23A, No. 3.

Evans, L., R. Herman, and T. Lam (1976). MultivariateAnalysis of Traffic Factors Related to Fuel Consumption inUrban Driving. Transportation Science, Vol. 10, No. 2, pp.205-215.

Evans, L. and R. Herman (1978). Automobile Fuel Economyon Fixed Urban Driving Schedules. Transportation Science,Vol. 12, No. 2, pp. 137-152.

Everall, P. F. (1968). The Effect of Road and TrafficConditions on Fuel Consumption. RRL Report LR 226,Road Research Laboratory, Crowthorne, England.

Concentrations Near Roadways. Report FHWA/CA/TL -

Computer Package for Simulating Multi-Lane TrafficFlows. Proceedings of the 4th Biennial Conference ofSimulation Society of Australia.

Greene, D. L. and J. T. Liu (1988). Automotive Fuel EconomyImprovements and Consumers' Surplus. TransportationResearch, Vol. 22A, No. 3.

Greene, D. L. and K. G. Duleep (1993). Costs and Benefits ofAutomotive Fuel Economy Improvement: A PartialAnalysis. Transportation Research, Vol. 27A, No.3.

Hauer, E. (1971). Accident Overtaking and Speed Control.

Hauer, E. (1992). Empirical Bayes Approach To TheEstimation Of "Unsafety"; The Multivariate RegressionMethod. Accident Analysis & Prev., Vol. 24, No. 5,pp. 457-477.

Hauer, E., D. Terry, and M. S. Griffith (1994). The Effect OfResurfacing On The Safety Of Rural Roads In New YorkState. Paper 940541 presented at the 73rd Annual Meetingof the Transportation Research Board, Washington, DC.

Hall, J. W. And O. J. Pendleton (1990). Rural Accident RateVariation with Traffic Volume. Transportation ResearchRecord 1281, TRB, NRC, Washington, DC, pp. 62-70.

Herman, R. and S. Ardekani (1985). The Influence of Stops onVehicle Fuel Consumption in Urban Traffic. TransportationScience, Vol. 19, No. 1, pp. 1-12.

Hickman, A. J. and V. H. Waterfield (1984). A User'sGuide to the Computer Programs for Predicting AirPollution from Road Traffic. TRRL Supplementary Report806.

Hickman, A. J. and M. H. L. Waters (1991). ImprovingAutomobile Fuel Economy. Traffic Engineering & Control,Vol. 23, No. 11.

Hooker, J. N., A. B. Rose, and G. F. Roberts (1983). OptimalControl of Automobiles for Fuel Economy. TransportationScience, Vol. 17, No. 2, pp. 146-167.

Institute of Transportation Engineers (1989). A Toolbox forAlleviating Traffic Congestion. Publication No. IR.054A.

��

� � ��

Jovanis, P. P. and H-L. Chang (1987). Modelling TheRelationship Of Accidents To Miles Travelled.Transportation Research Record 1068, Washington, DC, Document For Automotive Fuel Economy. Thepp. 42-51. Transportation Energy Management Program (TEMP)

Kenworthy, J. R., H. Rainford, P. W. G. Newman, and T. J. Report No. DRS-82-01, Ontario, Canada.Lyons (1986). Fuel Consumption, Time Saving andFreeway Speed Limits. Traffic Engineering & Control, Vol.27, No. 9, pp. 455-459.

Kihlberg, J. K. and K. J. Tharp (1968). Accident RatesAs Related To Design Elements Of Rural Highways.National Cooperative Highway Research Program Report,47, Highway Research Board.

Komor, P., S. F. Baldwin, and J. Dunkereley (1993). Pfundt, K. (1968). Comparative Studies Of AccidentsTechnologies for Improving Transportation Energy On Rural Roads. Strassenbau und Strassenverkehrstechnik,Efficiency in the Developing World. TransportationResearch, Vol 27A, No. 5. Pfundt, K. (1969). Three Difficulties In The Comparison Of

Lam, T. N. (July 1985). Estimating Fuel Consumption Accident Rates. Accident Analysis & Prevention, Vol. 1, pp.From Engine Size. Journal of Transportation Engineering,Vol. 111, No. 4, pp. 339-357.

Leutzbach, W., W. Siegener, and R. Wiedemann (1970). J. R. Kenworthy (1987). Fuel Consumption Models: AnÜber den Zusammenhang Zwischen Verkehrsunfällen Und Evaluation Based on a Study of Perth's Traffic Patterns .Verkehrsbelastung Auf Eienem deutschenAutobahnabschnitt (About the Relationship between TrafficAccidents and Traffic Flow on a Section of a GermanFreeway). Accident Analysis & Prev. Emissions Research Annual Report. Charles KollingVol. 2, pp. 93-102. Research Laboratory Technical Note: ER36, University of

Lieberman, E., R. D. Worrall, D. Wicks, and J. Woo (1979). Sydney.NETSIM Model (5 Vols.). Federal Highway Administration, Quaye, K., L. Leden, and E. Hauer (1993). PedestrianReport No. FHWA-RD-77-41 to 77-45, Washington, DC.

Lundy, R. A. (1965). Effect of Traffic Volumes And NumberOf Lanes On Freeway Accident Rates. Highway Research Washington, DC.Record 99, HRB, National Research Council, Washington,DC, pp. 138-147.

Mahalel, D. (1986). A Note on Accident Risk. TransportationResearch Record, 1068, National Research Council,Washington, DC, pp. 85-89.

McDonald, J. W. (1953). Relationship between Number ofAccidents and Traffic Volume at Divided-HighwayIntersections. Bulletin No. 74, pp. 7-17.

Miaou, S-P. and H. Lum (1993). Modeling Vehicle AccidentsAnd Highway Geometric Design Relationships. AccidentAnalysis & Prev. Vol. 25, No. 6, pp. 689-709.

Ng, J. C. N. and E. Hauer (1989). Accidents on RuralTwo-Lane Roads: Differences Between Seven States.Transportation Research Record, 1238,pp. 1-9.

Ontario Ministry of Transportation andCommunications (1982). A Technical Background

Pelensky, E., W. R. Blunden, and R. D. Munro (1968). Operating Costs of Cars in Urban Areas. Proceedings,Fourth Conference of the Australian Road Research Board,Vol. 4, part 1, pp. 475-504.

Persaud B. and Dzbik, L. (1993). Accident Prediction ModelsFor Freeways. Transportation Research Record 1401, TRB,NRC, Washington, DC, pp. 55-60.

Vol. 82, Federal Ministry of Transport, Bonn.

253-259.Pitt, D. R., T. J. Lyons, P. W. G. Newman, and

Traffic Engineering & Control, Vol. 27, No. 2, pp. 60-68.Post, K., J. Tomlin, D. Pitt, N. Carruthers, A. Maunder,

J. H. Kent, and R. W. Bilger (1981). Fuel Economy and

Accidents And Left-turning Traffic At SignalizedIntersections. AAA foundation for Traffic Safety,

Richards, G. G. (1983). MICRO2 - An Air QualityIntersection Model. Colorado Department of Highways.

Richardson, A. J. and R. Akcelik (1983). Fuel Consumptionand Data Needs for the Design and Evaluation of UrbanTraffic Systems. Research Report ARR No. 124, AustrailianRoad Research Board, pp. 51-56.

Roosmark, P.O. (1966). Trafikolyckor i trevagskorsningar.Statens Vaginstitut, Preliminary Report 22, Stockholm.

Satterthwaite, S. P. (1981). A Survey into Relationshipsbetween Traffic Accidents and Traffic Volumes. Transportand Road Research Laboratory Supplementary Report, SR692, pp. 41.

Slatterly, G. T. and D. E. Cleveland (1969). Traffic Volume. Traffic Control And Roadway Elements. Chapter 2,Automotive Safety Foundation, pp. 8.

��

� � ��

Tanner, J. C. (1953). Accidents At Rural Three-wayJunctions. J. Instn. of Highway Engrs. II(11), pp. 56-67.

Thorson, O. (1967). Traffic Accidents and Road Layout.Copenhagen Technical University of Denmark.

Transportation Research Board (1981). Methodology forEvaluating Highway Air Pollution Dispersion Models.NCHRP Report No. 245, National Research Council. Simplified Method for Quantifying Fuel Consumption of

U.S. DOT (1990). A Statement of National Transportation Vehicles in Urban Traffic. SAE-Australia, Vol. 40, No. 1,Policy, Strategies for Action, Moving America - NewDirections, New Opportunities.

Urban Mass Transportation Administration (1985). and Traffic Volumes at Signalized Intersections. Institute ofCharacteristics of Urban Transportation System. U.S.Department of Transportation. Welbourne, E. R. (1979). Exposure Effects on the Rate

U. S. EPA (1990). National Air Quality and Emissions and Severity of Highway Collisions at Different TravelTrends Report. Office of Air Quality. Speeds. Technical Memorandum TMVS 7903, Vehicle

U. S. EPA (1991). MOBILE4.1 User's Guide - MobileSource Emission Factor Model. Report EPA-AA-TEB-91-01.

Vincent, R. A., A. I. Mitchell, and D. I. Robertson (1980). Two-Lane Roads. Volume I, Final Report, FHWA-RD-User Guide to TRANSYT Version 8. Transport and RoadResearch Lab Report No. LR888.

Wallace, C. E., K. G. Courage, D. P. Reaves, G. W. Schoene,and G. W. Euler (1984). TRANSY-7F User's Manual. U.S.Dept. of Transportation, Office of Traffic Operations.

Walsh, M. P. (1989). Statement Before the U.S. SenateEnvironment and Public Works Committee.

Watson, H. C., E. E. Milkins, and G. A. Marshal (1980). A

pp. 6-13.Webb, G. M. (1955). The Relationship Between Accidents

Traffic Engineers. Proceedings.

Systems, Transport Canada.Zegeer, C. V., J. Hummer, D. Reinfurt, L. Herf, and W.

Hunter (1986). Safety Effects of Cross-Section Design for

87/008, Federal Highway Administration, TransportationResearch Board.

Professor, Head of the School, Civil Engineering, Queensland University of Technology, 2 George Street,13

Brisbane 4000 Australia.

Professor, Institute for Transportation, Faculty of Civil Engineering, Ruhr University, D 44780 Bochum,14

Germany.

UNSIGNALIZED INTERSECTION THEORY

BY ROD J. TROUTBECK13

WERNER BRILON14


b = proportion of volume of movement i of the total volume on the shared lanei

C = coefficient of variation of service timesw

D = total delay of minor street vehiclesD = average delay of vehicles in the queue at higher positions than the firstq

E(h) = mean headwayE(t ) = the mean of the critical gap, tc c

f(t) = density function for the distribution of gaps in the major streamg(t) = number of minor stream vehicles which can enter into a major stream gap of size, tL = logarithmm = number of movements on the shared lanen = number of vehicles� = increment, which tends to 0, when Var(t ) approaches 0c c

� = increment, which tends to 0, when Var(t ) approaches 0f f

q = flow in veh/secq = capacity of the shared lane in veh/hs

q = capacity of movement i, if it operates on a separate lane in veh/hm,i

q = the entry capacitym

q = maximum traffic volume departing from the stop line in the minor stream in veh/secm

q = major stream volume in veh/secp

t = timet = critical gap timec

t = follow-up timesf

t = the shift in the curvem

Var(t ) = variance of critical gapsc

Var(t ) = variance of follow-up-timesf

Var (W) = variance of service timesW = average service time. It is the average time a minor street vehicle spends in the first position of the queue near the intersectionW = service time for vehicles entering the empty system, i.e no vehicle is queuing on the vehicle's arrival1

W = service time for vehicles joining the queue when other vehicles are already queuing2

8 - 1

8.UNSIGNALIZED INTERSECTION THEORY

8.1 IntroductionUnsignalized intersections are the most common intersection driver and the pattern of the inter-arrival times aretype. Although their capacities may be lower than other important.intersection types, they do play an important part in the controlof traffic in a network. A poorly operating unsignalized This chapter describes both of these aspects when there are twointersection may affect a signalized network or the operation of streams. The theory is then extended to intersections with morean Intelligent Transportation System. than two streams.

The theory of the operation of unsignalized intersections isfundamental to many elements of the theory used for otherintersections. For instance, queuing theory in traffic engineeringused to analyze unsignalized intersections is also used to analyzeother intersection types.

8.1.1 The Attributes of a GapAcceptance Analysis Procedure

Unsignalized intersections give no positive indication or controlto the driver. He or she is not told when to leave the intersection.The driver alone must decide when it is safe to enter theintersection. The driver looks for a safe opportunity or "gap" inthe traffic to enter the intersection. This technique has beendescribed as gap acceptance. Gaps are measured in time and areequal to headways. At unsignalized intersections a driver mustalso respect the priority of other drivers. There may be othervehicles that will have priority over the driver trying to enter thetraffic stream and the driver must yield to these drivers.

All analysis procedures have relied on gap acceptance theory tosome extent or they have understood that the theory is the basisfor the operation even if they have not used the theory explicitly.

Although gap acceptance is generally well understood, it isuseful to consider the gap acceptance process as one that has twobasic elements.

� First is the extent drivers find the gaps or opportunitiesof a particular size useful when attempting to enter theintersection.

� Second is the manner in which gaps of a particular size aremade available to the driver. Consequently, the proportionof gaps of a particular size that are offered to the entering

8.1.2 Interaction of Streams atUnsignalized Intersections

A third requirement at unsignalized intersections is that theinteraction between streams be recognized and respected. At allunsignalized intersections there is a hierarchy of streams. Somestreams have absolute priority, while others have to yield tohigher order streams. In some cases, streams have to yield tosome streams which in turn have to yield to others. It is usefulto consider the streams as having different levels of priority orranking. For instance:

Rank 1 stream - has absolute priority and does not need toyield right of way to another stream,

Rank 2 stream - has to yield to a rank 1 stream,

Rank 3 stream - has to yield to a rank 2 stream and in turn toa rank 1 stream, and

Rank 4 stream - has to yield to a rank 3 stream and in turn toa rank 2 stream and to a rank 1 stream.

8.1.3 Chapter Outline

Sections 8.2 discusses gap acceptance theory and this leads toSection 8.3 which discusses some of the common headwaydistributions used in the theory of unsignalized intersections.

Most unsignalized intersections have more than two interactingstreams. Roundabouts and some merges are the only examplesof two interacting streams. Nevertheless, an understanding ofthe operation of two streams provides a basis to extend theknowledge to intersections with more than two streams. Section


8 - 2

8.4 discusses the performance of intersections with two The theory described in this chapter is influenced by the humaninteracting streams. factors and characteristics as described in Chapter 3, and in

Section 8.5 to 8.8 discuss the operation of more complex similarities between the material in this chapter and Chapter 9intersections. Section 8.9 covers other theoretical treatments of dealing with signalized intersections. Finally, unsignalizedunsignalized intersections. In many cases, empirical approaches intersections can quickly become very complicated and often thehave been used. For instance the relationships for AWSC (All subject of simulation programs. The comments in Chapter 10Way Stop Controlled) intersections are empirical. The time are particularly relevant here.between successive departures of vehicles on the subject roadway are related to the traffic conditions on the other roadwayelements.

particular, Sections 3.13 and 3.15. The reader will also note

8.2 Gap Acceptance Theory

8.2.1 Usefulness of Gaps

The gap acceptance theory commonly used in the analysis ofunsignalized intersections is based on the concept of defining theextent drivers will be able to utilize a gap of particular size orduration. For instance, will drivers be able to leave the stop lineat a minor road if the time between successive vehicles from theleft is 10 seconds; and, perhaps how many drivers will be ableto depart in this 10 second interval ?

The minimum gap that all drivers in the minor stream areassumed to accept at all similar locations is the critical gap.According to the driver behavior model usually assumed, nodriver will enter the intersection unless the gap between vehiclesin a higher priority stream (with a lower rank number) is at leastequal to the critical gap, t . For example, if the critical gap wasc4 seconds, a driver would require a 4 second gap between Rank1 stream vehicles before departing. He or she will require thesame 4 seconds at all other times he or she approaches the sameintersection and so will all other drivers at that intersection.

Within gap acceptance theory, it is further assumed that anumber of drivers will be able to enter the intersection from aminor road in very long gaps. Usually, the minor streamvehicles (those yielding right of way) enter in the long gaps atheadways often referred to as the "follow-up time", t .f

Note that other researchers have used a different concept for thecritical gap and the follow-up time. McDonald and Armitage(1978) and Siegloch (1973) independently described a conceptwhere a lost time is subtracted from each major stream gap and

the remaining time is considered 'useable.' This 'useable' timedivided by the saturation flow gives an estimate of the absorptioncapacity of the minor stream. As shown below, the effect of thisdifferent concept is negligible.

In the theory used in most guides for unsignalized intersectionsaround the world, it is assumed that drivers are both consistentand homogeneous. A consistent driver is expected to behave thesame way every time at all similar situations. He or she is notexpected to reject a gap and then subsequently accept a smallergap. For a homogeneous population, all drivers are expected tobehave in exactly the same way. It is, of course, unreasonable toexpect drivers to be consistent and homogeneous.

The assumptions of drivers being both consistent andhomogeneous for either approach are clearly not realistic.Catchpole and Plank (1986), Plank and Catchpole (1984),Troutbeck (1988), and Wegmann (1991) have indicated that ifdrivers were heterogeneous, then the entry capacity would bedecreased. However, if drivers are inconsistent then the capacitywould be increased. If drivers are assumed to be both consistentand homogeneous, rather than more realistically inconsistent andheterogeneous, then the difference in the predictions is only afew percent. That is, the overall effect of assuming that driversare consistent and homogeneous is minimal and, for simplicity,consistent and homogeneous driver behavior is assumed.

It has been found that the gap acceptance parameters t and tc fmay be affected by the speed of the major stream traffic (Harders1976 and Troutbeck 1988). It also expected that drivers areinfluenced by the difficulty of the maneuver. The more difficult

tf


8 - 3

a maneuver is, the longer are the critical gap and follow-up time distribution of follow-up times and the critical gap distributionparameters. There has also been a suggestion that drivers independently. Each group is discussed below.require a different critical gap when crossing different streamswithin the one maneuver. For instance a turn movement acrossa number of different streams may require a driver having adifferent critical gap or time period between vehicles in eachstream (Fisk 1989). This is seen as a unnecessary complicationgiven the other variables to be considered.

8.2.2 Estimation of the Critical Gap Parameters

The two critical gap parameters that need to be estimated are thecritical gap t and the follow-up time . The techniques used tocestimate these parameters fit into essentially two differentgroups. The first group of techniques are based on a regressionanalysis of the number of drivers that accept a gap againstthe gap size. The other group of techniques estimates the

Regression techniques.If there is a continuous queue on the minor street, then thetechnique proposed by Siegloch (1973) produces acceptableresults because the output matches the assumptions used in acritical gap analysis. For this technique, the queue must have atleast one vehicle in it over the observation period. The processis then:

� Record the size of each gap, t, and the number ofvehicles, n, that enter during this gap;

� For each of the gaps that were accepted by only ndrivers, calculate the average gap size, E(t) (See Figure8.1);

� Use linear regression on the average gap size values(as the dependent variable) against the number ofvehicles that enter during this average gap size, n; and

Figure 8.1Data Used to Evaluate Critical Gaps and Move-Up Times

(Brilon and Grossmann 1991).

tc � to � tf /2


8 - 4

(8.1)

� Given the slope is t and the intercept of the gap sizef

axis is t , then the critical gap t is given by o c

The regression line is very similar to the stepped line as shownin Figure 8.2. The stepped line reflects the assumptions made byTanner (1962), Harders (1976), Troutbeck (1986), and others.The sloped line reflects the assumptions made by Siegloch(1973), and McDonald and Armitage (1978).

Independent assessment of the critical gap and follow-up timeIf the minor stream does not continuously queue, then theregression approach cannot be used. A probabilistic approachmust be used instead.

The follow-up time is the mean headway between queuedvehicles which move through the intersection during the longergaps in the major stream. Consider the example of two majorstream vehicles passing by an unsignalized intersection at times2.0 and 42.0 seconds. If there is a queue of say 20 vehicleswishing to make a right turn from the side street, and if 17 of

these minor street vehicles depart at 3.99, 6.22, 8.29, 11.13,13.14, and so on, then the headways between the minor streetvehicles are 6.22-3.99, 8.29-6.22, 11.13-8.29 and so on. Theaverage headway between this group of minor streamvehicles is 2.33 sec. This process is repeated for a number oflarger major stream gaps and an overall average headwaybetween the queued minor stream vehicles is estimated. Thisaverage headway is the follow-up time, t . If a minor streamfvehicle was not in a queue then the preceding headway wouldnot be included. This quantity is similar to the saturationheadway at signalized intersections.

The estimation of the critical gap is more difficult. There havebeen numerous techniques proposed (Miller 1972; Ramsey andRoutledge 1973; Troutbeck 1975; Hewitt 1983; Hewitt 1985).The difficulty with the estimation of the critical gap is that itcannot be directly measured. All that is known is that a driver'sindividual critical gap is greater than the largest gap rejected andshorter than the accepted gap for that driver. If the accepted gapwas shorter than the largest rejected gap then the driver isconsidered to be inattentive. This data is changed to a value justbelow the accepted gap. Miller (1972) gives an alternativemethod of handling this inconsistent data which uses the data asrecorded. The difference in outcomes is generally marginal.

Figure 8.2Regression Line Types.

�n

i�1[F(ai)�F(ri)]

L � �n

i�1ln[F(ai)�F(ri)]

�L�μ

� 0

�L��

2� 0

�F(x)�μ

� �f(x)

�F(x)��

2� �

x�μ2�2

f(x)


8 - 5

(8.2)

(8.3)

(8.4)

(8.5)

(8.6)

(8.7)

Miller (1972), and later Troutbeck (1975) in a more limitedstudy, used a simulation technique to evaluate a total of tendifferent methods to estimate the critical gap distribution ofdrivers. In this study the critical gaps for 100 drivers weredefined from a known distribution. The arrival times of prioritytraffic were simulated and the appropriate actions of the"simulated" drivers were noted. This process was repeated for100 different sets of priority road headways, but with the sameset of 100 drivers. The information recorded included the sizeof any rejected gaps and the size of the accepted gap and wouldbe similar to the information able to be collected by an engineerat the road side. The gap information was then analyzed usingeach of the ten different methods to give an estimate of theaverage of the mean of the drivers' critical gaps, the variance ofthe mean of the drivers' critical gaps, mean of the standarddeviation of the drivers' critical gaps and the variance of thestandard deviation of the drivers' critical gaps. These statisticsenabled the possible bias in predicting the mean and standarddeviation of the critical gaps to be estimated. Techniques whichgave large variances of the estimates of the mean and thestandard deviation of the critical gaps were considered to be lessreliable and these techniques were identified. This procedurefound that one of the better methods is the Maximum LikelihoodMethod and the simple Ashworth (1968) correction to theprohibit analysis being a strong alternative. Both methods aredocumented here. The Probit or Logit techniques are alsoacceptable, particularly for estimating the probability that a gapwill be accepted (Abou-Henaidy et al. 1994), but more careneeds to be taken to properly account for flows. Kyte et al(1996) has extended the analysis and has found that theMaximum Likelihood Method and the Hewitt (1983) modelsgave the best performance for a wide range of minor stream andmajor stream flows.

The maximum likelihood method of estimating the critical gaprequires that the user assumes a probabilistic distribution of thecritical gap values for the population of drivers. A log-normal isa convenient distribution. It is skewed to the right and does nothave non-negative values. Using the notation:

a = the logarithm of the gap accepted by the ith driver,i

a = � if no gap was accepted,i

r = the logarithm of the largest gap rejected by the ithidriver,

r = 0 if no gap was rejected,i

μ and� are the mean and variance of the logarithm of the2

individual drivers critical gaps (assuming a log-normal distribution), and

f( ) and F( ) are the probability density function and the

cumulative distribution function respectively forthe normal distribution.

The probability that an individual driver's critical gap will bebetween r and a is F(a ) – F(r ). Summing over all drivers, thei i i i

likelihood of a sample of n drivers having accepted and largestrejected gaps of (a , r ) isi i

The logarithm, L, of this likelihood is then

The maximum likelihood estimators, μ and � , that maximize L,2

are given by the solution to the following equations.

and

Using a little algebra,

�n

i�1

f(ri)� f(ai)F(ai)�F(ri)

� 0

�n

i�1

(ri�μ ) f(ri)�(ai�μ ) f(ai)F(ai) � F(ri)

� 0

E(tc) � e μ�0.5�2

Var(tc) � E(tc )2 (e �2�1)

E(tc) � E(ta)�qp Var(ta)

μ


8 - 6

(8.8)

(8.9)

(8.10)

(8.11)

(8.12)

This then leads to the following two equations which must be first gap offered without rejecting any gaps, then Equations 8.8solved iteratively. It is recommended that the equation and 8.9 give trivial results. The user should then look at

should be used to estimate μ given a value of � . An initial value2

of � is the variance of all the a and r values. Using this2i i

estimate of μ from Equation 8.8, a better estimate of � can be2

obtained from the equation,

where is an estimate of μ.

A better estimate of the μ can then be obtained from theEquation 8.8 and the process continued until successiveestimates of μ and � do not change appreciably.2

The mean, E(t ), and the variance, Var(t ), of the critical gap normal function is used, then E(t ) and Var(t ) are values givenc cdistribution is a function of the log normal distribution by the generic Equations 8.10 and 8.11. This is a very practicalparameters, viz: solution and one which can be used to give acceptable results in

and

The critical gap used in the gap acceptance calculations is thenequal to E(t ). The value should be less than the mean of thecaccepted gaps.

This technique is a complicated one, but it does produceacceptable results. It uses the maximum amount of information,without biasing the result, by including the effects of a largenumber of rejected gaps. It also accounts for the effects due tothe major stream headway distribution. If traffic flows werelight, then many drivers would accept longer gaps withoutrejecting gaps. On the other hand, if the flow were heavy, allminor stream drivers would accept shorter gaps. Thedistribution of accepted gaps is then dependent on the majorstream flow. The maximum likelihood technique can account forthese different conditions. Unfortunately, if all drivers accept the

alternative methods or preferably collect more data.

Another very useful technique for estimating the critical gap isAshworth’s (1968) procedure. This requires that the useridentify the characteristics of the probability distribution thatrelates the proportion of gaps of a particular size that wereaccepted to the gap size. This is usually done using a Probitanalysis applied to the recorded proportions of accepted gaps.A plot of the proportions against the gap size on probabilitypaper would also be acceptable. Again a log normal distributionmay be used and this would require the proportions to be plottedagainst the natural logarithm of the gap size. If the mean andvariance of this distribution are E(t ) and Var(t ), thena aAshworth’s technique gives the critical gap as

where q is the major stream flow in units of veh/sec. If the logp

a a

the office or the field.

8.2.3 Distribution of Gap Sizes

The distribution of gaps between the vehicles in the differentstreams has a major effect on the performance of theunsignalized intersection. However, it is important only to lookat the distribution of the larger gaps; those that are likely to beaccepted. As the shorter gaps are expected to be rejected, thereis little point in modeling these gaps in great detail.

A common model uses a random vehicle arrival pattern, that is,the inter-arrival times follow an exponential distribution. Thisdistribution will predict a large number of headways less than 1sec. This is known to be unrealistic, but it is used because thesesmall gaps will all be rejected.

This exponential distribution is known to be deficient at highflows and a displaced exponential distribution is oftenrecommended. This model assumes that vehicle headways areat least t sec.m

P(n) � (qt)n e �qt

n!

P(h>t) � e �qt

P(h�t) � 1�e �qt

f(t) � d[P(h�t)]dt

� q e �qt

q�[P(h>0.1)]�3600


8 - 7

(8.13)

(8.14)

(8.15)

(8.16)

Better models use a dichotomized distribution. These models In this chapter, the word "queues" is used to refer to a line ofassume that there is a proportion of vehicles that are free of stopped vehicles. On the other hand, a platoon is a group ofinteractions and travel at headways greater than t . These traveling vehicles which are separated by a short headway of t .mvehicles are termed "free" and the proportion of free vehicles is When describing the length of a platoon, it is usual to include a�. There is a probability function for the headways of free platoon leader which will have a longer headway in front of himvehicles. The remaining vehicles travel in platoons and again or her. A platoon of length one is a single vehicle travellingthere is a headway distribution for these bunched vehicles. One without any vehicles close-by. It is often useful to distinguishsuch dichotomized headway model is Cowan's (1975) M3 model between free vehicles (or platoon leaders) and those vehicles inwhich assumes that a proportion, �, of all vehicles are free and the platoon but behind the leader. This latter group are calledhave an displaced exponential headway distribution and the 1-� bunched vehicles. The benefits of a number of different headwaybunched vehicles have the same headway of only t .m

m

models will be discussed later.

8.3 Headway Distributions Used in Gap Acceptance Calculations8.3.1 Exponential Headways

The most common distribution is the negative exponentialdistribution which is sometimes referred to as simply the"exponential distribution". This distribution is based on theassumption that vehicles arrive at random without anydependence on the time the previous vehicle arrived. Thedistribution can be derived from assuming that the probability ofa vehicle arriving in a small time interval (t, t+�t) is a constant.It can also be derived from the Poisson distribution which givesthe probability of n vehicles arriving in time t, that is:

where q is the flow in veh/sec. For n = 0 this equation gives theprobability that no vehicle arrives in time t. The headway, h,must be then greater than t and the probability, from Equation If the flow was 1440 veh/h or 0.4 veh/sec then the number of8.13 is headways less than 0.1 seconds is then or

The cumulative probability function of headways is then

The probability distribution function is then be considered to be the space around a vehicle that no other

This is the equation for the negative exponential distribution.The parameter q can be estimated from the flow or the reciprocalof the average headway. As an example, if there were 228headways observed in half an hour, then the flow is 228/1800 i.e.q = 0.127 veh/sec. The proportion of headways expected to begreater than 5 seconds is then

P(h>5) = e –q t

= e – 5*0.127

= 0.531

The expected number of headways greater than 5 secondsobserved in half an hour is then 0.531�228 or 116.

56 per hour. This over-estimation of the number of very shortheadways is considered to be unrealistic and the displacedexponential distribution is often used instead of the negativeexponential distribution.

8.3.2 Displaced Exponential Distribution

The shifted or displaced exponential distribution assumes thatthere is a minimum headway between vehicles, t . This time canm

F(h) � 1�e ��(h�tm)

� �

q1�tmq

E(h) � 1/q� tm �

1�

p(h�t) � 1��e �t/hf�(1��)e �(t�tm)/(hb�tm)

hf

hb

p(h � t) � 1 � �e ��(t�tm) for t > tm

� �

�q(1�tmq)

� � e �Aqp


8 - 8

(8.17)

(8.18)

(8.19)

(8.20)

(8.21)

(8.22)

(8.23)

vehicle can intrude divided by the traffic speed. If the flow is qveh/h then in one hour q vehicles will pass and there are t � qmseconds lost while these vehicles pass. The remaining time mustthen be distributed randomly after each vehicle and the averagerandom component is (1-t � q)/q seconds. The cumulativemprobability distribution of headways is then:

where,

There, the terms, � and t need to be evaluated. These can bemestimated from the mean and the variance of the distribution.The mean headway, E(h), is given by:

The variance of headways is 1/� . These two relationships can2

then be used to estimate � and t .m

This distribution is conceptually better than the negativeexponential distribution but it does not account for theplatooning that can occur in a stream with higher flows. Adichotomized headway distribution provides a better fit.

8.3.3 Dichotomized Headway Distributions

In most traffic streams there are two types of vehicles, the firstare bunched vehicles; these are closely following precedingvehicles. The second group are free vehicles that are travellingwithout interacting with the vehicles ahead. There have been anumber of dichotomized headway distributions developed overtime. For instance, Schuhl (1955) proposed a distribution

where there are � vehicles that are free (not in platoons);

there are (1–�) bunched vehicles;

is the average headway for free vehicles;

is the average headway for bunched or constrainedvehicles;

t is the shift in the curve.m

Other composite headway models have been proposed byBuckley (1962; 1968). However, a better headway model forgap acceptance is the M3 model proposed by Cowan (1975).This model does not attempt to model the headways between thebunched vehicles as these are usually not accepted but rathermodels the larger gaps. This headway model has a cumulativeprobability distribution:

andp(h� t) = 0 otherwise.

Where � is a decay constant given by the equation

Cowan's headway model is rather general. To obtain thedisplaced exponential distribution set � to 1.0. For the negativeexponential distribution, set � to 1.0 and t to 0. Cowan's modelmcan also give the headway distribution used by Tanner (1962) bysetting � to 1–t q, however the distribution of the number ofmvehicles in platoons is not the same. This is documented below.

Brilon (1988) indicated that the proportion of free vehicles couldbe estimated using the equation,

where A values ranged from 6 to 9. Sullivan and Troutbeck(1993) found that this equation gave a good fit to data, frommore than 600 of hours of data giving in excess of 400,000vehicle headways, on arterial roads in Australia. They alsofound that the A values were different for different lanes and fordifferent lane widths. These values are listed in Table 8.1.

p(h�t) � 1��e �(t�tmf / hf�tmf )

� (1��)e �(t�tmb/hb�tmb)�k�1

x�0

kt�tmb

hb� tmb

x

x!

p(h � t) � 1�e �0.0465t


8 - 9

(8.24)

Table 8.1 theory when predicting capacity or delays. The hyper-Erlang“A” Values for Equation 8.23 from Sullivan andTroutbeck (1993).

Median All otherLane lanes

Lane width < 3.0 meters 7.5 6.5

3.0 � Lane width � 3.5 meters 7.5 5.25

Lane width> 3.5 meters 7.5 3.7

Typical values of the proportion of free vehicles are given inFigure 8.3.

The hyper-Erlang distribution is also a dichotomized headwaydistribution that provides an excellent fit to headway data. It isuseful in simulation programs but has not been used in traffic

distribution given by Dawson (1969) is:

8.3.4 Fitting the Different HeadwayModels to Data

If the mean headway is 21.5 seconds and standard deviation is19.55 seconds, then the flow is 1/21.5 or 0.0465 veh/seconds(167 veh/hour). A negative exponential curve that would fit thisdata is then,

Figure 8.3Typical Values for the Proportion of Free Vehicles.

p(h�t) � 1�e �0.0512(t�1.94)

P(n) � (1��)n�1�

n �

1�

Var(n) � 1��

2


8 - 10

(8.25)

(8.26)

(8.27)

To estimate the parameters for the displaced exponentialdistribution, the difference between the mean and the standarddeviation is the displacement, that is t is equal to 21.49 – 19.55mor 1.94 seconds. The constant � used in Equation 8.21 is thereciprocal of the standard deviation. In this case, � is equal to1/19.55 or 0.0512 veh/sec. The appropriate equation is then:

The data and these equations are shown in Figure 8.4 whichindicates the form of these distributions. The reader should notmake any conclusions about the suitability of a distribution fromthis figure but should rather test the appropriateness of the modelto the data collected.

In many cases there are a substantial number of very shortheadways and a dichotomized headway distribution performsbetter. As only the larger gaps are likely to be accepted bydrivers, there is no point in modeling the shorter gaps in greatdetail. An example of Cowan’s M3 model and headway datafrom an arterial road is shown in Figure 8.5. Figure 8.6 gives time of t . Although the distribution of these 'revised' majorthe same data and the hyper-Erlang distribution.

Under these conditions the mean platoon size is

and the variance by

Another distribution of platoons used in the analysis ofunsignalized intersections is the Borel-Tanner distribution. Thisplatooning distribution comes from Tanner's (1962) assumptionswhere the major stream gaps are the outcome of a queuingprocess with random arrivals and a minimum inter-departure

m

stream gaps is given by Equation 8.21 with � equal to 1–t q,m

Figure 8.4Exponential and Displaced Exponential Curves

(Low flows example).


8 - 11

Figure 8.5Arterial Road Data and a Cowan (1975) Dichotomized Headway Distribution

(Higher flows example).

Figure 8.6 Arterial Road Data and a Hyper-Erlang Dichotomized Headway Distribution

(Higher Flow Example).

P(n) �e �ntmq (ntmq)n�1

n!

qm � qp ��

0f(t) �g(t)dt

tm q / (1 – tmq )3


8 - 12

(8.28)

(8.29)

the distribution of the platoon length is Borel-Tanner (Borel Haight and Breuer (1960) found the mean platoon size to be 1942; Tanner 1953; 1961; and Haight and Breuer 1960). Again,q is the flow in veh/sec. The Borel-Tanner distribution of (1- �) /� . For the same mean platoon size, the Borel-Tannerplatoons gives the probability of a platoon of size n as

where n is an integer.

1 / (1 - t q ) or 1/� and the variance to be orm3

distribution has a larger variance and predicts a greater numberof longer platoons than does the geometric distribution.Differences in the platoon size distribution does not affect anestimate of capacity but it does affect the average delay pervehicle as shown in Figure 8.13.

8.4 Interaction of Two Streams

For an easy understanding of traffic operations at an unsignalizedintersection it is useful to concentrate on the simplest case first(Figure 8.7).

All methods of traffic analysis for unsignalized intersections arederived from a simple queuing model in which the crossing oftwo traffic streams is considered. A priority traffic stream(major stream) of the volume q (veh/h) and a non-priority trafficp

stream (minor stream) of the volume q (veh/h) are involved innthis queuing model. Vehicles from the major stream can crossthe conflict area without any delay. Vehicles from the minor where,stream are only allowed to enter the conflict area, if the nextvehicle from the major stream is still t seconds away (t is the stop line in the minor stream in veh/sec,c ccritical gap), otherwise they have to wait. Moreover, vehiclesfrom the minor stream can only enter the intersection t secondsf

after the departure of the previous vehicle (t is the follow-up the major stream, andftime).

8.4.1 Capacity

The mathematical derivation of the capacity q for the minorm

stream is as follows. Let g(t) be the number of minor streamvehicles which can enter into a major stream gap of duration t.The expected number of these t-gaps per hour is 3600q f(t) pwhere,

f(t) = statistical density function of the gaps in themajor stream and

q = volume of the major stream.p

Therefore, the amount of capacity which is provided by t-gapsper hour is 3600 q f(t) g(t).p

To get the total capacity, expressed in veh/second, we have tointegrate over the whole range of major stream gaps:

q = maximum traffic volume departing from them

q = major stream volume in veh/sec,p

f(t) = density function for the distribution of gaps in

g(t) = number of minor stream vehicles which canenter into a major stream gap of size, t .

Based on the gap acceptance model, the capacity of the simple2-stream situation (Figure 8.7) can be evaluated by elementaryprobability theory methods if we assume:

(a) constant t and t values,c f

(b) exponential distribution for priority stream headways(cf. Equation 8.15), and

(c) constant traffic volumes for each traffic stream.

g(t) � ��

n�0n�pn(t)

pn(t) �1 for tc�(n�1)�tf�t<tc�n�tf0 elsewhere

g(t) �0 for t < t0

t�t0tf

for t � t0


8 - 13

(8.30)

(8.31)

Figure 8.7Illustration of the Basic Queuing System.

Within assumption (a), we have to distinguish between twodifferent formulations for the term g(t). These are the reason fortwo different families of capacity equations. The first familyassumes a stepwise constant function for g(t) (Figure 8.2):

where,p (t)= probability that n minor streamn

vehicles enter a gap in the major streamof duration t,

The second family of capacity equations assumes a continuouslinear function for g(t) . This is an approach which has first beenused by Siegloch (1973) and later also by McDonald andArmitage (1978).

t0 � tc�tf

2


8 - 14

where,

Once again it has to be emphasized that both in Equations 8.30and 8.31, t and t are assumed to be constant values for all different manner:c fdrivers.

Both approaches for g(t) produce useful capacity formulae wherethe resulting differences are rather small and can normally beignored for practical applications (cf. Figure 8.8).

If we combine Equations 8.29 and 8.30, we get the capacityequation used by Drew (1968), Major and Buckley (1962), andby Harders (1968), which these authors however, derived in a

Figure 8.8Comparison Relation Between Capacity (q-m) and Priority Street Volume (q-p).

qm � qpe �qp�tc

1�e �qp�t f

qm �

1tf

e �qp�tc

tf


8 - 15

(8.32)

(8.33)

If we combine Equations 8.29 and 8.31 we get Siegloch's (1973)formula,

These formulae result in a relation of capacity versus conflictingflow illustrated by the curves shown in Figure 8.8.

The idealized assumptions, mentioned above as (a), (b), (c),however, are not realistic. Therefore, different attempts to dropone or the other assumption have been made. Siegloch (1973)studied different types of gap distributions for the priority stream(cf. Figure 8.9) based on analytical methods. Similar studieshave also been performed by Catchpole and Plank (1986) andTroutbeck (1986). Grossmann (1991) investigated these effectsby simulations. These studies showed

� If the constant t and values are replaced by realisticcdistributions (cf. Grossmann 1988) we get a decreasein capacity.

� Drivers may be inconsistent; i.e. one driver can havedifferent critical gaps at different times; A driver mightreject a gap that he may otherwise find acceptable. Thiseffect results in an increase of capacity.

� If the exponential distribution of major stream gaps isreplaced by more realistic headway distributions, weget an increase in capacity of about the same order ofmagnitude as the effect of using a distribution for t andc

t values (Grossmann 1991 and Troutbeck 1986).f

� Many unsignalized intersections have complicateddriver behavior patterns, and there is often little to begained from using a distribution for the variables t andc

t or complicated headway distributions. Moreover,fGrossmann could show by simulation techniques thatthese effects compensate each other so that the simplecapacity equations, 8.32 and 8.33, also give quiterealistic results in practice.

Note: Comparison of capacities for different types of headway distributions in the main street traffic flow for t = 6 seconds andc

t = 3 seconds. For this example, t has been set to 2 seconds.f m

Figure 8.9Comparison of Capacities for Different Types of

Headway Distributions in the Main Street Traffic Flow.

qm �

�qpe��(tc�tm)

1�e ��t f

� ��qf

(1�tmqf )

qm � (1�qp�tm)�qp�e

�qp�(tc�tm)

1�e �qp�tf

qm �

��qp�e��(t0�tm)

��tf

qm �

(1�qptm)�e ��(t0�tm)

tf

qp� tm


8 - 16

(8.34)

(8.35)

(8.36)

(8.37)

(8.38)

More general solutions have been obtained by replacing the equation:exponential headway distribution used in assumption (b) with amore realistic one e. g. a dichotomized distribution (cf. Section8.3.3). This more general equation is:

where

This equation is illustrated in Figure 8.10. This is also similarto equations reported by Tanner (1967), Gipps (1982),Troutbeck (1986), Cowan (1987), and others. If � is set to 1and t to 0, then Harders' equation is obtained. If � is set toml– , then this equation reduces to Tanner's (1962) This was proposed by Jacobs (1979) .

If the linear relationship for g(t) according to Equation 8.37 isused, then the associated capacity equation is

or

Figure 8.10The Effect of Changing �� in Equation 8.31 and Tanner's Equation 8.36.

qm � ��

qp�e�qp�tc

1�e �qp�tf

� � 1� 12

�q 2p Var(tc )�

Var(tf )

(e qp�tf�1)

��c��f

qm �

1�E z(G�tc)/�E(C)�E(1/�)

tc


8 - 17

(8.39)

(8.40)

(8.41)Tanner (1962) analyzed the capacity and delay at an intersectionwhere both the major and minor stream vehicles arrived atrandom; that is, their headways had a negative exponentialdistribution. He then assumed that the major stream vehicleswere restrained such that they passed through the intersection atintervals not less than t sec after the preceding major streammvehicle. This allowed vehicles to have a finite length into which where,other vehicles could not intrude. Tanner did not apply the sameconstraint to headways in the minor stream. He assumed thesame gap acceptability assumptions that are outlined above.Tanner considered the major stream as imposing 'blocks' and'anti-blocks' on the minor stream. A block contains one or moreconsecutive gaps less than t sec; the block starts at the firstc

vehicle with a gap of more than t sec in-front of it and ends c

sec after the last consecutive gap less than t sec. Tanner'scequation for the entry capacity is a particular case of a moregeneral equation.

An analytical solution for a realistic replacement of assumptions(a) and (b) within the same set of formulae is given by Plank andCatchpole (1984):

where

Var(t ) = variance of critical gaps,c

Var(t ) = variance of follow-up-times,f

� = increment, which tends to 0, when Var(tc c

) approaches 0, and� = increment, which tends to 0, when Var(tf f

) approaches 0.

Wegmann (1991) developed a universal capacity formula whichcould be used for each type of distribution for the critical gap, forthe follow-up time and for each type of the major streamheadway distribution.

E(C) = mean length of a "major road cycle" C,C = G + B,G = gap,B = block,� = probability (G > t ), andc

z(t) = expected number of departures withinthe time interval of duration t.

Since these types of solutions are complicated many researchershave tried to find realistic capacity estimations by simulationstudies. This applies especially for the German method (FGSV1991) and the Polish method.

8.4.2 Quality of Traffic Operations

In general, the performance of traffic operations at anintersection can be represented by these variables (measures ofeffectiveness, MOE):

(a) average delay,(b) average queue lengths,(c) distribution of delays,(d) distribution of queue lengths (i.e number of vehicles

queuing on the minor road),(e) number of stopped vehicles and number of

accelerations from stop to normal velocity, and(f) probability of the empty system (p ).o

Distributions can be represented by:

� standard deviations,� percentiles, and� the whole distribution.

To evaluate these measures, two tools can be used to solve theproblems of gap acceptance:

� queuing theory and� simulation.

D � Dmin 1��x1�x

� �e qptf

�qptf �1�qp(eqptf�1)Dmin

qp(eqptf�1)Dmin

Dmin �e �(tc�tm)

�qp

� tc �1��

�t 2m�2tm�2tm�

2(tm��)

Dmin �e qp(tc�tm)

(1�tmqp)qp� tc �

1qp�

qpt2m(2tmqp�1)

2(1�tmqp)2

D �

1�e �(qptc�qntf)

qm/3600�qn�tf

Dq�xW(1�C 2

w)2(1�x)


8 - 18

(8.42)

(8.43)

(8.44)

(8.45)

(8.46)

(8.47)

Each of these MOEs are a function of q and q ; the proportionp nof "free" vehicles and the distribution of platoon size length inboth the minor and major streams. Solutions from queuingtheory in the first step concentrate on average delays.

A general form of the equation for the average delay per vehicleis

where � and � are constantsx is the degree of saturation = q /qn m

and D has been termed Adams' delay after Adams (1936).minAdams' delay is the average delay to minor stream vehicles whenminor stream flow is very low. It is also the minimum averagedelay experienced by minor stream vehicles.

Troutbeck (1990) gives equations for �, � and D based on theminformulations by Cowan (1987). If stream 2 vehicles areassumed to arrive at random, then � is equal to 0. On the otherhand, if there is platooning in the minor stream, then � is greaterthan 0.

For random stream 2 arrivals, � is given by

Note that � is approximately equal to 1.0. D depends on theminplatooning characteristics in stream 1. If the platoon sizedistribution is geometric, then

(Troutbeck 1986).

Tanner's (1962) model has a different equation for Adams' delay,because the platoon size distribution in stream 1 has a Borel-Tanner distribution. This equation is

Another solution for average delay has been given by Harders(1968). It is not based on a completely sophisticated queuingtheory. However, as a first approximation, the followingequation for the average delay to non-priority vehicles is quiteuseful.

with q calculated using Equation. 8.34 or similar.m

M/G/1 Queuing System - A more sophisticated queuing theorymodel can be developed by the assumption that the simple two-streams system (Figure 8.7) can be represented by a M/G/1queue. The service counter is the first queuing position on theminor street. The input into the system is formed by the vehiclesapproaching from the minor street which are assumed to arriveat random, i.e. exponentially distributed arrival headways (i.e."M"). The time spent in the first position of the queue is theservice time. This service time is controlled by the prioritystream, with an unknown service time distribution. The "G" isfor a general service time. Finally, the "1" in M/G/1 stands forone service channel, i.e. one lane in the minor street.

For the M/G/1 queuing system, in general, the Pollaczek-Khintchine formula is valid for the average delay of customers inthe queue

whereW = average service time. It is the average

time a minor street vehicle spends in thefirst position of the queue near theintersection

C = coefficient of variation of service timesw

Cw �

Var(W)W

D �

1qm

1� x1�x

C

C �

1�C 2w

2

D�Dmin (1��) 1� ��1��

�

x1�x

Dq�qn

2�

E(W 21 )�E(W2

2 )v

�

E(W22 )

y

tf


8 - 19

(8.48)

(8.49)

(8.51)

Var (W) = variance of service times

The total average delay of minor street vehicles is then

D = D + W.q

In general, the average service time for a single-channel queuingsystem is: l/capacity. If we derive capacity from Equations 8.32 critical gap parameters t and and the headway distributions.and following and if we include the service time W in the total However, C, �, and � values are not available for all conditions.delay, we get

where

Up to this point, the derivations are of general validity. The realproblem now is to evaluate C. Only the extremes can be definedwhich are:

� Regular service: Each vehicle spends the same time in thefirst position. This gives Var(W) = 0, C = 0, and C = w

2

0.5

This is the solution for the M/D/l queue.

� Random service: The times vehicles spend in the firstposition are exponentially distributed. This gives Var(W) = E(W), C = 1, and C = 1.0w

2

This gives the solution for the M/M/1 queue.

Unfortunately, neither of these simple solutions applies exactlyto the unsignalized intersection problem. However, as anapproximation, some authors recommend the application ofEquation 8.48 with C = 1.

Equation 8.42 can be further transformed to

where � and � are documented in Troutbeck (1990).

This is similar to the Pollaczek-Khintchine formula (Equation8.48). The randomness constant C is given by (�+�)/(l+�) andthe term 1/D *(l+�) can be considered to be an equivalentmin'capacity' or 'service rate.' Both terms are a function of the

c

For the M/G/1 system as a general property, the probability p ofothe empty queue is given by

p = 1 - x (8.50)o

This formula is of sufficient reality for practical use atunsignalized intersections.

M/G2/1 queuing system - Different authors found that theservice time distribution in the queuing system is betterdescribed by two types of service times, each of which has aspecific distribution:

W = service time for vehicles entering the empty system, i.e1no vehicle is queuing on the vehicle's arrival

W = service time for vehicles joining the queue when other2vehicles are already queuing.

Again, in both cases, the service time is the time the vehiclespends waiting in the first position near the stop line. The firstideas for this solution have been introduced by Kremser (1962;1964) and in a comparable way by Tanner (1962), as well as byYeo and Weesakul (1964).

The average time which a customer spends in the queue of sucha system is given by Yeo's (1962) formula:

where,

D�E(W2

1 )v

�

qn

2�

y�E(W21 )�z�E(W 2

2 )v�y

E(W1) �1qp

(e qpt

E(W22 ) � 2�e qptc

q 2p

(e qptc�qptc)(1�e �qptf)�qptf�e

�

E(W 21 ) � 2

qp

(e qptc�1�qptc)(

e qptc

qp

�tf�tc)�t 2f �t 2

c

E(W2) �e qptc

qp

(1�e �qptf)


8 - 20

(8.53)

(8.54)

D = average delay of vehicles in the queue atqhigher positions than the first,

E(W ) = expectation of W ,1 1

E(W ) = expectation of (W *W )1 1 12

E(W ) = expectation of W ,2 2

E(W ) = expectation of (W * W ),2 2 22

v = y + z,y = 1 –q E(W ), andn 2

z = q E(W ).n l

The probability p of the empty queue iso

p = y/v (8.52)o

The application of this formula shows that the differences againstEquation 8.50 are quite small ( < 0.03). Refer to Figure 8.11.

If we also include the service time ( = time of minor streetvehicles spent in the first position) in the total delay, we get

(Brilon 1988):Formulae for the expectations of W and W1 2respectively have been developed by Kremser (1962):

Figure 8.11Probability of an Empty Queue: Comparison of Equations 8.50 and 8.52.


8 - 21

Kremser (1964), however, showed that the validity of these regarded as approximations and only apply forequations is restricted to the special case of t = t , which isc frather unrealistic for two-way-stop-control unsignalizedintersections. Daganzo (1977) gave an improved solution forE(W ) and E(W ) which again was extended by Poeschl (1983).2 2

2

These new formulae were able to overcome Kremer's (1964)restrictions. It can, however be shown that Kremer's firstapproach (Equation 8.56) also gives quite reliable approximateresults for t and t values which apply to realistic unsignalizedc fintersections. The following comments can also be made aboutthe newer equations.

� The formulae are so complicated that they are far frombeing suitable for practice. The only imaginable applicationis the use in computer programs.

� Moreover, these formulae are only valid under assumptions(a), (b), and (c) in Section 8.4.1 of the paper. That meansthat for practical purposes, the equations can only be

undersaturated conditions and steady state conditions.

Figure 8.12 gives a graphical comparison for some of the delayformulae mentioned.

Differences in the platoon size distribution affects the averagedelay per vehicle as shown in Figure 8.13. Here, the critical gapwas 4 seconds, the follow-up time was 2 seconds, and thepriority stream flow was 1000 veh/h. To emphasize the point,the average delay for a displaced exponential priority stream is4120 seconds, when the minor stream flow was 400 veh/h. Thisis much greater than the values for the Tanner and exponentialheadway examples which were around 11.5 seconds for the samemajor stream flow. The average delay is also dependent on theaverage platoon size as shown in Figure 8.14. The differencesin delays are dramatically different when the platoon size ischanged.

Note: For this example; q = 600 veh/h, t = 6 sec , and t = 3 sec.p c f

Figure 8.12Comparison of Some Delay Formulae.


8 - 22

Figure 8.13Average Steady State Delay per Vehicle

Calculated Using Different Headway Distributions.

Figure 8.14Average Steady State Delay per Vehicle by Geometric

Platoon Size Distribution and Different Mean Platoon Sizes.

p(1) � p(0)�h3�qn e qntf�(tc�tf)�h2 �qn�h1�h3

p(0) � h1�h3�(qp�qn)

p(n) � p(n�1)�h3�qn e qntf�(tc�tf)�h2

�h3 ��n�2

m�0p(m) � h2

(tc� tf � qn)n�m

(n�m)!�

(�qntf )n�m�e qntf

tf �(n�m�1)!

h1 � e �qptc�(e �qptf

�1)qn

qp

h2 � qpe�qptc�qn(tc�tf)

1h3

� h2�qn�e�qptf

p(0) � 1�x a

p(n) � p(0)�x a(b(n�1)�1)

x �

qn

qm

a �

1

1 � 0.45 �

tc�tf

tf� qp

b �

1.51

1� 0.68 �

tc

tf� qp

a �

11 � 0.45 � qp

b �

1.511�1.36�qp

F(n) � p(L�n) � 1�x a(b �n�1)


8 - 23

(8.57)

(8.58)

(8.59)

8.4.3 Queue Length

In each of these queuing theory approaches, the average queuelength (L) can be calculated by Little's rule (Little 1961):

L = q D (8.55)n

Given that the proportion of time that a queue exists is equal tothe degree of saturation, the average queue length when there isa queue is:

L = q D/x = q D (8.56)q n m

The distribution of queue length then is often assumed to begeometric.

However, a more reliable derivation of the queue lengthdistribution was given by Heidemann (1991). The followingversion contains a correction of the printing mistakes in theoriginal paper (there: Equations 8.30 and 8.31).

p(n) = probability that n vehicles are queuing on the minor street

These expressions are based on assumptions (a), (b), and (c) inSection 8.4.1. This solution is too complicated for practical use.Moreover, specific percentiles of the queue length is the desiredoutput rather than probabilities. This however, can not be

calculated from these equations directly. Therefore, Wu (1994)developed another set of formulae which approximate the abovementioned exact equations very closely:

p(n) = probability that n vehicles are queuing on the minor street

where,

x = degree of saturation (q according to Equationm8.33).

For the rather realistic approximation t � 2 t , we get :c f

From Equation 8.58 we get the cumulative distribution function

For a given percentile, S, (e.g. S = F(n) = 0.95) this equationcan be solved for n to calculate the queue length which is onlyexceeded during (1-S)*100 percent of the time (Figure 8.15).For practical purposes, queue length can be calculated withsufficient precision using the approximation of the M/M/1queuing system and, hence, Wu’s equation. The 95-percentile-queue length based on Equation 8.59 is given in Figure 8.15.

P(x,0) � 1� (1�x)(1�tmqp)e��(tc�tm)

P(x,t) � P(0,t)�A�1�P(0,t)�x�(1�A)�1�P(0,t)�x 2

�(1�A)(1�B)(1�x)x

B � 1� (1� ttf

)(1�tmqp)e��(ta�tm)

A � 1�a0e��(ta�tm)

P(0,t) � P(0,0)� qpt�e ��(ta�tm)


8 - 24

(8.60)

(8.61)

(8.62)

The parameter of the curves (indicated on the right side) is the degree of saturation ( x ).

Figure 8.1595-Percentile Queue Length Based on Equation 8.59 (Wu 1994).

8.4.4 Stop Rate

The proportion of drivers that are stopped at an unsignalizedintersection with two streams was established by Troutbeck(1993). The minor stream vehicles were assumed to arrive atrandom whereas the major stream headways were assumed tohave a Cowan (1975) M3 distribution. Changes of speed areassumed to be instantaneous and the predicted number ofstopped vehicles will include those drivers who could haveadjusted their speed and avoided stopping for very short periods.

The proportion stopped, P(x,0), is dependent upon the degree ofsaturation, x, the headways between the bunched major streamvehicles, t , the critical gap, t . and the major stream flow, q .m c pThe appropriate equation is:

where � is given by �q /(1-t q ). The proportion of driversp m p

stopped for more than a short period of t, where t is less than the

follow-up time t , increases from some minimum value, P(0,t),fto 1 as the degree of saturation increases from 0 to 1.

The proportion of drivers stopped for more than a short periodt, P(x,t), is given by the empirical equation:

where

and

P(0,t) � 1� (1�tmqp�qpt�)e ��(ta�tm)

T > 1

qm� qn2


8 - 25

(8.63)

or

If the major stream is random then a is equal to 1.25 and for0bunched major stream traffic, it is 1.15. The vehicles that arestopped for a short period may be able to adjust their speed andthese vehicles have been considered to have a “partial stop."Troutbeck (1993) also developed estimates of the number oftimes vehicles need to accelerate and move up within the queue.

8.4.5 Time Dependent Solution

Each of the solutions given by the conventional queuing theoryabove is a steady state solution. These are the solutions that canbe expected for non-time-dependent traffic volumes after aninfinitely long time, and they are only applicable when the degreeof saturation x is less than 1. In practical terms, this means,the results of steady state queuing theory are only useful

approximations if T is considerably greater than the expressionon the right side of the following equation.

with T = time of observation over which the average delayshould be estimated in seconds,

after Morse (1962).

This inequality can only be applied if q and q are nearlym n

constant during time interval T. The threshold given byEquation 8.63 is illustrated by Figure 8.16. The curves are givenfor time intervals T of 5, 10, 15, 30, and 60 minutes. Steadystate conditions can be assumed if q is below the curve for then

corresponding T-value. If this condition (Equation 8.63) is notfulfilled, time-dependent solutions should be used.Mathematical solutions for the time dependent problem havebeen developed by Newell (1982) and now need to be made

Note: The curves are given for time intervals T of 5, 10, 15, 30, and 60 minutes. Steady state cond itions can be assumed if q isn

below the curve for the corresponding T-value.

Figure 8.16Approximate Threshold of the Length of Time Intervals For the Distinction

Between Steady-State Conditions and Time Dependent Situations.

D � D1�E� 1qm

D1 �12

F 2�G�F

F �

1qmo�qno

T2

(qm�qn)y�C(y� hqm

) �E

G �

2Tyqmo�qno

Cqn

qm�(qm�qn)E

E �

Cqno

qmo(qmo�qno)h � qm�qmo�qno

y � 1� hqn

Dd � Dmin�2L0�(xd�1)qmT

2qm

xs�Ds�Dmin��Dmin

Ds�Dmin��Dmin

Dd Ds


8 - 26

(8.64)

(8.67)

more accessible to practicing engineers. There is, however, a This delay formula has proven to be quite useful to estimateheuristic approximate solution for the case of the peak hour delays and it has a quite reliable background particularly foreffect given by Kimber and Hollis (1979) which are based on the temporarily oversaturated conditions.ideas of Whiting, who never published his work.

During the peak period itself, traffic volumes are greater than ordinate transfer method. This is a more approximate method.those before and after that period. They may even exceed The steady state solution is fine for sites with a low degree ofcapacity. For this situation, the average delay during the peak saturation and the deterministic solution is satisfactory for sitesperiod can be estimated as: with a very high degree of saturation say, greater than three or

q = capacity of the intersection entry during them

peak period of duration T,q = capacity of the intersection entry before andmo

after the peak period,q = minor street volume during the peak period ofn

duration T, andq = minor street volume before and after the peakno

period

(each of these terms in veh/sec; delay in sec).

C is again similar to the factor C mentioned for the M/G/1system, where

C = 1 for unsignalized intersections andC = 0.5 for signalized intersections (Kimber and Hollis

1979).

A simpler equation can be obtained by using the same co-

four. The co-ordinate transfer method is one technique toprovide estimates between these two extremes. the readershould also refer to Section 9.4.

The steady state solution for the average delay to the enteringvehicle is given by Equation 8.42. The deterministic equationfor delay, D , on the other hand isd

x > 1 (8.65)

and D = 0dotherwise,

where L is the initial queue,0

T is time the system is operating in seconds, andq is the entry capacity.m

These equations are diagrammatically illustrated in Figure 8.17.For a given average delay the co-ordinate transformation methodgives a new degree of saturation, x , which is related to thet

steady state degree of saturation, x , and the deterministic degrees

of saturation, x , such thatd

x – x = 1 – x = a (8.66)d t s

Rearranging Equations 8.42 and 8.65 gives two equations for xs

and x as a function of the delays and . These twodequations are:

xd �2(Dd�Dmin)�2L0/qm

T� 1

xt �2(Dd�Dmin)�2L0/qm

T�

Ds�Dmin��Dmin

Ds�Dmin��Dmin

Dt �12

A 2�B�A

A �

T(1�x)2

�

L0

qm� Dmin(2��)

B � 4DminT(1�x)(1��)

2�

Tx(��)2

�(1��)L0

qm�Dmin


8 - 27

(8.68)

(8.69)

(8.70)

(8.71)

(8.72)

Figure 8.17The Co-ordinate Transform Technique.

and

Using Equation 8.66, x is given by:t

Rearranging Equation 8.69 and setting D = D = D , x = x gives:s d �

where

and

Equation 8.66 ensures that the transformed equation willasymptote to the deterministic equation and gives a family ofrelationships for different degrees of saturation and period ofoperation from this technique (Figure 8.18).

A simpler equation was developed by Akçelik in Akcelik andTroutbeck (1991). The approach here is to rearrange Equation8.42 to give:

a � 1�xs �Dmin(��xs)

Ds�Dmin

a �

Dmin(��xt)Ds�Dmin

D�Dmin �12

L0

qm

�

(x�1)T4

�

L0

2qm�

(x�1)T4

2�

TDmin(�x��)2

D �

1qm

�

T4

(x�1)� (x�1)2�

8xqmT

R � qemax�qn


8 - 28

(8.73)

(8.74)

(8.75)

(8.76)

Figure 8.18A Family of Curves Produced from the Co-Ordinate Transform Technique.

and this is approximately equal to: The average delay predicted by Equation 8.74 is dependent on

If this is used in Equation 8.66 and then rearranged then theresulting equation of the non-steady state delay is:

A similar equation for M/M/1 queuing system can be obtainedif � is set to 1, � is set to zero, and D is set to 1/q ; the resultmin mis:

the initial queue length, the time of operation, the degree ofsaturation, and the steady state equation coefficients. Thisequation can then be used to estimate the average delay underoversaturated conditions and for different initial queues. The useof these and other equations are discussed below.

8.4.6 Reserve Capacity

Independent of the model used to estimate average delays, thereserve capacity (R) plays an important role

b� 1qm�Rf

L0�RfT2

1�Rf

R0�

L0

qm

1�Rf�

D��B� B 2� b

B� 12

bR�L0

qm

L0�qn0

R0�

qm0�R0

R0

Rf�100 � 3600

T


8 - 29

(8.79)

(8.77)

(8.78)

(8.81)

(8.80)

In the 1985 edition of the HCM but not the 1994 HCM, it is usedas the measure of effectiveness. This is based on the fact thataverage delays are closely linked to reserve capacity. This closerelationship is shown in Figure 8.19. In Figure 8.19, the averagedelay, D, is shown in relation to reserve capacity, R. The delaycalculations are based on Equation 8.64 with a peak hourinterval of duration T= 1 hour. The parameters (100, 500, and1000 veh/hour) indicate the traffic volume, q , on the majorpstreet. Based on this relationship, a good approximation of theaverage delay can also be expressed by reserve capacities. Whatwe also see is that - as a practical guide - a reserve capacity

R > 100 pcu/h generally ensures an average delaybelow 35 seconds.

Brilon (1995) has used a coordinate transform technique for the"Reserve Capacity" formulation for average delay withoversaturated conditions. His set of equations can be given by

where

T = duration of the peak periodq = capacity during the peak periodm

q = minor street flow during the peak periodn

R = reserve capacity during the peak period = q – qemax n

L = average queue length in the period before and0after the peak period

q = minor street flow in the period before and aftern0the peak period

q = capacity in the period before and after the peakm0period

R = reserve capacity in the period before and after the0peak period

Figure 8.19Average Delay, D, in Relation to Reserve Capacity R.

qemax �3600qe �qta

1�e �qtf


8 - 30

(8.82)

All variables in these equations should be used in units of depart according to the gap acceptance mechanism. Theseconds (sec), number of vehicles(veh), and veh/sec. Any effect of limited acceleration and deceleration can, ofcapacity formula to estimate q and q from Section 8.4.1 canm m0be used.

The numerical results of these equations as well as their degreeof sophistication are comparable with those of Equation 8.75.

8.4.7 Stochastic Simulation

As mentioned in the previous chapters, analytical methods arenot capable of providing a practical solution, given thecomplexity and the assumptions required to be made to analyzeunsignalized intersections in a completely realistic manner. Themodern tool of stochastic simulation, however, is able toovercome all the problems very easily. The degree of reality ofthe model can be increased to any desired level. It is onlyrestricted by the efforts one is willing to undertake and by theavailable (and tolerable) computer time. Therefore, stochasticsimulation models for unsignalized intersections were developedvery early (Steierwald 1961a and b; Boehm, 1968). More recentsolutions were developed in the U. K. (Salter 1982), Germany(Zhang 1988; Grossmann 1988; Grossmann 1991), Canada(Chan and Teply 1991) and Poland (Tracz 1991).

Speaking about stochastic simulation, we have to distinguish twolevels of complexity:

1) Point Process Models - Here cars are treated like points,i.e. the length is neglected. As well, there is only limiteduse of deceleration and acceleration. Cars are regarded asif they were "stored" at the stop line. From here they

course, be taken into account using average vehicleperformance values (Grossmann 1988). The advantage ofthis type of simulation model is the rather shorter computertime needed to run the model for realistic applications.One such model is KNOSIMO (Grossmann 1988, 1991).It is capable of being operated by the traffic engineer onhis personal computer during the process of intersectiondesign. A recent study (Kyte et al , 1996) pointed out thatKNOSIMO provided the most realistic representation oftraffic flow at unsignalized intersections among a group ofother models.

KNOSIMO in its present concept is much related toGerman conditions. One of the specialities is therestriction to single-lane traffic flow for each direction ofthe main street. Chan and Teply (1991) found some easymodifications to adjust the model to Canadian conditionsas well. Moreover, the source code of the model couldeasily be adjusted to traffic conditions and driver behaviorin other countries.

2) Car Tracing Models - These models give a detailedaccount of the space which cars occupy on a road togetherwith the car-following process but are time consuming torun. An example of this type of model is described byZhang (1988).

Both types of models are useful for research purposes. Themodels can be used to develop relationships which can then berepresented by regression lines or other empirical evaluationtechniques.

8.5 Interaction of Two or More Streams in the Priority Road

The models discussed above have involved only two streams; a single lane with the opposing flow being equal to the sum ofone being the priority stream and the second being a minor the lane flows. This results in the following equation forstream. The minor stream is at a lower rank than the priority capacity in veh/h:stream. In some cases there may be a number of lanes that mustbe given way to by a minor stream driver. The capacity and thedelay incurred at these intersections have been looked at by anumber of researchers. A brief summary is given here.

If the headways in the major streams have a negative exponentialdistribution then the capacity is calculated from the equation for where q is the total opposing flow.

qemax �3600 �(1�tm1q1)(1�tm2q2)��(1�tm1q1)e

��(ta�tm)

1�e ��t f

qemax �

3600q�n

i�1(1�tmiqi)e

�qitai e qtm

1�e �qtf

F(t) �2q1q2t(q1�q2)

for t < tm

F(t) � 1��e ��(t�tm) for t > tm

��

�1q1(1�q2tm)��2q2(1�q1tm)(q1�q2)

��q � �

�

�n

i�1(1�qitm)

F(t) � 1�� e ��(t�t �m) t > t �m


8 - 31

(8.83)

(8.86)

(8.87)

(8.88)

(8.88a)

(8.88b)

(8.90)

Tanner (1967) developed an equation for the capacity of anintersection where there were n major streams. The traffic ineach lane has a dichotomized headway distribution in whichthere is a proportion of vehicles in bunches and the remainingvehicles free of interaction. All bunched vehicles are assumedto have a headway of t and the free vehicles have a headwaym

equal to the t plus a negative exponentially distributed (ormrandom) time. This is the same as Cowan's (1975) M3 model.Using the assumption that headways in each lane areindependent, Tanner reviewed the distribution of the randomtime periods and estimated the entry capacity in veh/h as:

where � = � + � + . . . . . + � (8.84)1 2 n

� = �q / (1-t q ) (8.85)i i i m i

q is the flow in the major stream i in veh/sec.i

� is the proportion of free vehicles in the major stream i.i

This equation by Tanner is more complicated than an earlierequation (Tanner 1962) based on an implied assumption that theproportion of free vehicles, � , is a function of the lane flow.iThat is

� = (1-t q )i m i

and then � reduces to q. Fisk (1989) extended this earlier worki iof Tanner (1962) by assuming that drivers had a different criticalgap when crossing different streams. While this would seem tobe an added complication it could be necessary if drivers arecrossing some major streams from the left before merging withanother stream from the right when making a left turn.

Her equation for capacity is:

where q = q + q + . . . . . + q1 2 n

8.5.1 The Benefit of Using aMulti-Lane Stream Model

Troutbeck (1986) calculated the capacity of a minor stream tocross two major streams which both have a Cowan (1975)dichotomized headway distribution. The distribution ofopposing headways is:

and

where

or after a little algebra,

and�' = � + � (8.89)1 2

As an example, if there were two identical streams then thedistribution of headways between vehicles in the two streams isgiven by Equations 8.87 and 8.88. This is also shown in figuresfrom Troutbeck (1991) and reported here as Figure 8.20.

Gap acceptance procedures only require that the longerheadways or gaps be accurately represented. The shorter gapsneed only be noted.

Consequently the headway distribution from two lanes can berepresented by a single Cowan M3 model with the followingproperties:

(1�t �mq1�t �mq2)e��t �m� (1�tmq1)(1�t �mq2)e

��tm

� � e ��t �m� �

�e ��tm

t �m,i�1 �

1� (1�tmq1)(1�tmq2)e��(tm�t �m,i)

q1�q2


8 - 32

(8.91)

(8.92)

(8.93)

Figure 8.20Modified 'Single Lane' Distribution of Headways (Troutbeck 1991).

and otherwise F(t) is zero. This modified distribution is alsoillustrated in Figure 8.20. Values of �* and t must be chosenm

*

to ensure the correct proportions and the correct mean headwayare obtained. This will ensure that the number of headwaysgreater than t, 1–F(t), is identical from either the one lane or thetwo lane equations when t is greater than t .m

*

Troutbeck (1991) gives the following equations for calculating�* and t which will allow the capacity to be calculated usingm

*

a modified single lane model which are identical to the estimatefrom a multi-lane model.

The equations

and

are best solved iteratively for t with t being the ith estimate.m m,iThe appropriate equation is

�* is then found from Equation 8.93.

Troutbeck (1991) also indicates that the error in calculatingAdams' delay when using the modified single lane model insteadof the two lane model is small. Adams' delay is the delay to theminor stream vehicles when the minor stream flow is close tozero. This is shown in Figure 8.21. Since the modifieddistribution gives satisfactory estimates of Adams' delay, it willalso give satisfactory estimates of delay.

In summary, there is no practical reason to increase thecomplexity of the calculations by using multi-lane models and asingle lane dichotomized headway model can be used torepresent the distribution of headways in either one or two lanes.


8 - 33

Figure 8.21Percentage Error in Estimating Adams' Delay Against the

Major Stream Flow for a Modified Single Lane Model (Troutbeck 1991).

8.6 Interaction of More than Two Streams of Different Ranking

8.6.1 Hierarchy of Traffic Streams at aTwo Way Stop Controlled Intersection

At all unsignalized intersections except roundabouts, there is ahierarchy of streams. Some streams have absolute priority(Rank 1), while others have to yield to higher order streams. Insome cases, streams have to yield to some streams which in turnhave to yield to others. It is useful to consider the streams ashaving different levels of priority or ranking. These differentlevels of priority are established by traffic rules. For instance,

Rank 1 stream has absolute priority and does not need toyield right of way to another stream,

Rank 2 stream has to yield to a Rank 1 stream,

Rank 3 stream has to yield to a Rank 2 stream and in turn toa Rank 1 stream, and

Rank 4 stream has to yield to a Rank 3 stream and in turn toRank 2 and Rank 1 streams (left turners fromthe minor street at a cross-intersection).

This is illustrated in Figure 8.22 produced for traffic on the rightside. The figure illustrates that the left turners on the major roadhave to yield to the through traffic on the major road. The leftturning traffic from the minor road has to yield to all otherstreams but is also affected by the queuing traffic in the Rank 2stream.

8.6.2 Capacity for Streams of Rank 3 and Rank 4

No rigorous analytical solution is known for the derivation of thecapacity of Rank-3-movements like the left-turner from theminor street at a T-junction (movement 7 in Figure 8.22, rightside). Here, the gap acceptance theory uses the impedancefactors p as an approximation. p for each movement is the0 0probability that no vehicle is queuing at the entry. This is givenwith sufficient accuracy by Equation 8.50 or better with the twoservice time Equation 8.52. Only during the part p of the0,rank-2total time, vehicles of Rank 3 can enter the intersection due tohighway code regulations.

pz,i � 0.65py,i�py,i

py,i�3�0.6 py,i


8 - 34

(8.97)

Note: The numbers beside the arrows indicate the enumeration of streams given by the Highway Capacity Manual (1994,Chapter 10).

Figure 8.22Traffic Streams And Their Level Of Ranking.

Therefore, for Rank-3-movements, the basic value q for them

potential capacity must be reduced to p � q to get the real0 m

potential capacity q :e

q = p q (8.94)e,rank-3 0,rank-2 m,rank-3.

For a T-junction, this means

q = p qe,7 0,4 m,7.

For a cross-junction, this means

q = p q (8.95)e,8 x m,8.

q = p q (8.96)e,11 x m,11.

with

p = p px 0,1 0,4.

Here the index numbers refer to the index of the movementsaccording to Figure 8.22. Now the values of p and p can be0,8 0,11calculated according to Equation 8.50.

For Rank-4-movements (left turners at a cross-intersection), thedependency between the p values in Rank-2 and Rank-3-0movements must be empirical and can not be calculated fromanalytical relations. They have been evaluated by numeroussimulations by Grossmann (1991; cf. Brilon and Grossmann1991). Figure 8.23 shows the statistical dependence betweenqueues in streams of Ranks 2 and 3.

In order to calculate the maximum capacity for the Rank-4-movements (numbers 7 and 10), the auxiliary factors, p andz,8

p , should be calculated first:z,11

diminished to calculate the actual capacities, q . Brilon (1988,ecf. Figures 8.7 and 8.8) has discussed arguments which supportthis double introduction.

The reasons for this are as follows:

1qs

� �m

i�1

bi

qm,i


8 - 35

(8.100)

Figure 8.23Reduction Factor to Account for the Statistical Dependence

Between Streams of Ranks 2 and 3.

� During times of queuing in Rank-2 streams (e.g. left � Even if no Rank-2 vehicle is queuing, these vehiclesturners from the major street), the Rank-3 vehicles (e.g. influence Rank-3 operations, since a Rank-2 vehicleleft turners from the minor street at a T-junction) cannotenter the intersection due to traffic regulations and thehighway code. Since the portion of time provided forRank-3 vehicles is p , the basic capacity calculated from Grossmann (1991) has proven that among the possibilities0Section 8.4.1 for Rank-3 streams has to be diminished by considered, the described approach is the easiest and quite the factor p for the corresponding Rank-2 streams0(Equations 8.95 to 8.99).

approaching the intersection within a time of less than tcprevents a Rank-3 vehicle from entering the intersection.

realistic in the range of traffic volumes which occur in practicalapplications.

8.7 Shared Lane Formula

8.7.1 Shared Lanes on the Minor Street

If more than one minor street movement is operating on the samelane, the so-called "shared lane equation" can be applied. Itcalculates the total capacity q of the shared lane, if the capacitiessof the corresponding movements are known. (Derivation inHarders, 1968 for example.)

q = capacity of the shared lane in veh/h,s

q = capacity of movement i, if it operates on am,iseparate lane in veh/h,

p �

o,i � 1�1�p0,i

1�qjtBj �qktBk

cT��

y k�1�1

y(y k�1) �[c(q5)�q1]�(y�1) � c(q1�q2�q5)

cT(y�1)��

k�1k[c(q5)�q1]�(c(q1�q2�q5)


8 - 36

(8.101)

(8.104)

b = proportion of volume of movement i of the totalivolume on the shared lane, those streams. The factors p * and p * indicate the probability

m = number of movements on the shared lane.

The formula is also used by the HCM (1994, Equation 10-9).

This equation is of general validity regardless of the formula forthe estimation of q and regardless of the rank of priority of themthree traffic movements. The formula can also be used if theoverall capacity of one traffic stream should be calculated, if thisstream is formed by several partial streams with differentcapacities, e.g. by passenger cars and trucks with differentcritical gaps. Kyte at al (1996) found that this procedure foraccounting for a hierarchy of streams, provided most realistic i = 4, j = 5 and k = 6 (cf. Figure 8.22)results.

8.7.2 Shared Lanes on the Major Street

In the case of a single lane on the major street shared by right-turning and through movements (movements no. 2 and 3 or 5and 6 in Figure 8.22), one can refer to Table 8.2.

If left turns from the major street (movements no. 1 and 4 inFigure 8.22) have no separate turning lanes, vehicles in thepriority l movements no. 2 and 3, and no. 5 and 6 respectively

in Figure 8.21 may also be obstructed by queuing vehicles in0,1 0,4

that there will be no queue in the respective shared lane. Theymight serve for a rough estimate of the disturbance that can beexpected and can be approximated as follows (Harders 1968):

where: i = 1, j = 2 and k = 3 (cf. Figure 8.22)

or

q = volume of stream j in veh/sec,j

q = volume of stream k in veh/sec, and k

t and t = follow-up time required by a vehicle in stream jBj Bk

or k (s).(1.7 sec < t < 2.5 sec, e.g. t = 2 sec)B B

In order to account for the influence of the queues in the majorstreet approach lanes on the minor street streams no. 7, 8, 10,and 11, the values p and p , according to Equation 8.47 have0,1 0,4

to be replaced by the values p * and p * according to Equation0,1 0,48.101. This replacement is defined in Equations 8.95 to 8.97.

8.8 Two-Stage Gap Acceptance and Priority

At many unsignalized intersections there is a space in the center the basis of an adjustment factor �. The resulting set ofof the major street available where several minor street vehicles equations for the capacity of a two-stage priority situation are:can be stored between the traffic flows of the two directions ofthe major street, especially in the case of multi-lane major traffic(Figure 8.24). This storage space within the intersection enablesthe minor street driver to cross the major streams from eachdirection at different behavior times. This behavior cancontribute to an increased capacity. This situation is called two-stage priority. The additional capacity being provided by thesewider intersections can not be evaluated by conventionalcapacity calculation models.

Brilon et al. (1996) have developed an analytical theory for theestimation of capacities under two-stage priority conditions. It isbased on an earlier approach by Harders (1968). In addition tothe analytical theory, simulations have been performed and were

for y � 1

for y = 1c = total capacity of the intersection for minor through trafficT

(movement 8)

y�c(q1�q2)�c (q1�q2�q5)c(q5)�q1�c (q1�q2�q5)

a�1–0.32exp(�1.3 � k)


8 - 37

Table 8.2Evaluation of Conflicting Traffic Volume qpNote: The indices refer to the traffic streams denoted in Figure 8.22.

Subject Movement No. Conflicting Traffic Volume qp

Left Turn from Major Road 1

7

q + q5 63)

q + q2 33)

Right Turn from Minor 6Road

12

q + 0.5 q2 32) 1)

q + 0.5 q5 62) 1)

Through Movement from 5Minor Road

11

q + 0.5 q + q + q + q + q2 3 5 6 1 41) 3)

q + q + q + 0.5 q ) + q + q2 3 5 6 1 43) 1

Left Turn from Minor Road 4

10

q + 0.5 q + q + q + q + q + q2 3 5 1 4 12 111) 4)5)6) 5)

q + 0.5 q + q + q + q + q + q5 6 2 1 4 6 81) 4)5)6) 5)

Notes1) If there is a right-turn lane, q or q should not be considered.3 62) If there is more than one lane on the major road, q and q are considered as traffic volumes on the right2 5

lane.3) If right-turning traffic from the major road is separated by a triangular island and has to comply with a

YIELD- or STOP-Sign, q and q need not be considered.6 34) If right-turning traffic from the minor road is separated by a triangular island and has to comply with a

YIELD- or STOP-sign, q and q need not be considered.9 125) If movements 11 and 12 are controlled by a STOP-sign, q and q should be halved in this equation.11 12

Similarly, if movements 8 and 9 are controlled by a STOP-sign, q and q should be halved.8 96) It can also be justified to omit q and q or to halve their values if the minor approach area is wide.9 12

where

q = volume of priority street left turning traffic at part I1

q = volume of major street through traffic coming from the2

left at part Iq = volume of the sum of all major street flows coming5

from the right at part II.

� = 1 for k=0

for k > 0 (8.105)

Of course, here the volumes of all priority movements at part IIhave to be included. These are: major right (6, except if thismovement is guided along a triangular island separated from thethrough traffic) , major through (5), major left (4); numbers ofmovements according to Figure 8.22.


8 - 38

Note: The theory is independent of the number of lanes in the major street.

Figure 8.24Minor Street Through Traffic (Movement 8) Crossing the Major Street in Two Phases.

c(q q ) = capacity at part I 1 + 2

c(q ) = capacity at part II5

c(q +q +q ) = capacity at a cross intersection for1 2 5minor through traffic with a majorstreet traffic volume of q +q +q1 2 5

(All of these capacity terms are to be calculated by any usefulcapacity formula, e.g. the Siegloch-formula, Equation 8.33)

The same set of formulas applies in analogy for movement 7. Ifboth movements 7 and 8 are operated on one lane then the totalcapacity of this lane has to be evaluated from c and c usingT7 T8the shared lane formula (Equation 8.95). Brilon et al. (1996)provide also a set of graphs for an easier application of thistheory.

8.9 All-Way Stop Controlled Intersections

8.9.1 Richardson’s Model

Richardson (1987) developed a model for all-way stopcontrolled intersections (AWSC) based on M/G/1 queuingtheory. He assumed that a driver approaching will either have

a service time equal to the follow-up headway for vehicles in thisapproach if there are no conflicting vehicles on the cross roads(to the left and right). The average service time is the timebetween successive approach stream vehicles being able todepart. If there were conflicting vehicles then the conflicting

sn �qwtmTc�tm�qwt 2

m

1�qwqn(T2c �2tmTc�t 2

m)

tc � 3.6 � 0.1 number of lanes

Ws �2��2

�q 2Var(s)2(1��)q

Ws ��

q1�

� 1� q 2

�2

Var(s)

2(1��)


8 - 39

(8.108)(8.116)

vehicles at the head of their queues will depart before theapproach stream being analysed. Consequently, Richardsonassumed that if there were conflicting vehicles then the averageservice time is the sum of the clearance time, T , for conflictingcvehicles and for the approach stream.

For simplicity, Richardson considered two streams; northboundand westbound. Looking at the northbound drivers, theprobability that there will be a conflicting vehicle on the crossroad is given by queuing theory as � . The average service timewfor northbound drivers is then

s = t (1–� ) + T � (8.106)n m w c w

A similar equation for the average service time for westbounddrivers is

s = t (1–� ) + T � (8.107)w m e c ewhere,

� is the utilization ratio and is q s ,i i i

q is the flow from approach i,i

s is the service time for approach ii

t is the minimum headway, andm

T is the total clearance time.c

These equations can be manipulated to give a solution for s asn

If there are four approaches then very similar equations areobtained for the average service time involving the probabilitythere are no cars on either conflicting stream. For instance,

s = t (1–� ) + T � (8.109)n m ew c ew

s = t (1–� ) + T � (8.110)s m ew c ew

s = t (1–� ) + T � (8.111)e m ns c ns

s = t (1–� ) + T � (8.112)w m ns c ns

The probability of no conflicting vehicles being 1–� given byns

1–� = (1–� ) (1–� ) (8.113)ns s n

hence,

� = 1 – (1–q s ) (1–q s ) (8.114)ns n n s s

and� = 1 – (1–q s ) (1–q s ) (8.115)ew e e w w

Given the flows, q , q , q , and q and using an estimate ofn s e w

service times, p and p can be estimated using Equationsns ew8.114 and 8.115. The iterative process is continued withEquations 8.109 to 8.112 providing a better estimate of theservice times, s , s , s , and s .n s e w

Richardson used Herbert’s (1963) results in which t was foundm

to be 4 sec and T was a function of the number of cross flowclanes to be crossed. The equation was

and T is the sum of the t values for the conflicting and thec capproach streams.

The steady-state average delay was calculated using thePollaczek-Khintchine formula with Little’s equation as:

or

This equation requires an estimate of the variance of the servicetimes. Here Richardson has assumed that drivers either had aservice time of h or T . For the northbound traffic, there werem c

(1–� ) proportion of drivers with a service time of exactly tew m

and � drivers with a service time of exactly T . The varianceew cis then

Var(s)n � t 2m(1��ew)�T 2

c ��s 2n

� �

sn�tm

Tc�tm

Var(s)n � t 2m

Tc�sn

Tc�tm� T 2

csn�tm

Tc�tm� s 2

n


8 - 40

(8.117)

(8.118)

(8.119)

and

This then gives

for the northbound traffic. Similar equations can be obtained forthe other approaches. An example of this technique applied to

a four way stop with single lane approaches is given in Figure8.25. Here the southbound traffic has been set to 300 veh/h.The east-west traffic varies but with equal flows in bothdirections. In accordance with the comments above, t was 4 secm

and T was 2*t or 7.6 sec. c c

Richardson's approach is satisfactory for heavy flows where mostdrivers have to queue before departing. His approach has beenextended by Horowitz (1993), who extended the number ofmaneuver types and then consequently the number of servicetime values. Horowitz has also related his model to Kyte’s(1989) approach and found that his modified Richardson modelcompared well with Kyte’s empirical data.

Figure 8.25 from Richardson's research, gives the performanceas the traffic in one set approaches (north-south or east-west)increases. Typically, as traffic flow in one directionincreases so does the traffic in the other directions. This willusually result in the level of delays increasing at a more rapidrate than the depicted in this figure.

Figure 8.25Average Delay For Vehicles on the Northbound Approach.


8 - 41

8.10 Empirical Methods

Empirical models often use regression techniques to quantify an or even other characteristic values of the intersection layout byelement of the performance of the intersection. These models, by another set of linear regression analysis (see e.g. Kimber andtheir very nature, will provide good predictions. However, at Coombe 1980).times they are not able to provide a cause and effect The advantages of the empirical regression technique comparedrelationships. to gap acceptance theory are:

Kimber and Coombe (1980), of the United Kingdom, have � there is no need to establish a theoretical model.evaluated the capacity of the simple 2-stream problem using � reported empirical capacities are used.empirical methods. The fundamental idea of this solution is as � influence of geometrical design can be taken into account.follows: Again, we look at the simple intersection (Figure 8.7) � effects of priority reversal and forced priority are taken intowith one priority traffic stream and one non-priority traffic account automatically.stream during times of a steady queue (i.e. at least one vehicle is � there is no need to describe driver behavior in detail.queuing on the minor street). During these times, the volume oftraffic departing from the stop line is the capacity. This capacity The disadvantages are:should depend on the priority traffic volume q during the sameptime period. To derive this relationship, observations of trafficoperations of the intersection have to be made during periods ofoversaturation of the intersection. The total time of observationthen is divided into periods of constant duration, e.g. 1 minute.During these 1-minute intervals, the number of vehicles in boththe priority flow and the entering minor street traffic are counted.Normally, these data points are scattered over a wide range andare represented by a linear regression line. On average, half ofthe variation of data points results from the use of one-minutecounting intervals. In practice, evaluation intervals of more than1-minute (e.g. 5-minutes) cannot be used, since this normallyleads to only few observations.

As a result, the method would produce linear relations for q :m

q = b - c q (8.120)m p.

Instead of a linear function, also other types of regression couldbe used as well, e.g.

q = A e . (8.121)m. -Bx

Here, the regression parameters A and B could be evaluated outof the data points by adequate regression techniques. This typeof equation is of the same form as Siegloch's capacity formula(Equation 8.33). This analogy shows that A=3600/t .f

In addition to the influence of priority stream traffic volumes onthe minor street capacity, the influence of geometric layout of theintersection can be investigated. To do this, the constant valuesb and c or A and B can be related to road widths or visibility

� transferability to other countries or other times (driverbehavior might change over time) is quite limited: Forapplication under different conditions, a very big samplesize must always be evaluated.

� no real understanding of traffic operations at the intersectionis achieved by the user.

� the equations for four-legged intersections with 12movements are too complicated.

� the derivations are based on driver behavior underoversaturated conditions.

� each situation to be described with the capacity formulaemust be observed in reality. On one hand, this requires alarge effort for data collection. On the other hand, many ofthe desired situations are found infrequently, sincecongested intersections have been often already signalized.

8.10.1 Kyte's Method

Kyte (1989) and Kyte et al. (1991) proposed another method forthe direct estimation of unsignalized intersection capacity forboth AWSC and TWSC intersections. The idea is based on thefact that the capacity of a single-channel queuing system is theinverse of the average service time. The service time, t , at theWunsignalized intersection is the time which a vehicle spends inthe first position of the queue. Therefore, only the average ofthese times (t ) has to be evaluated by observations to get theWcapacity.

Under oversaturated conditions with a steady queue on the minorstreet approach, each individual value of this time in the first

d��5

i�1P(Ci)hi

d

d d

d


8 - 42

position can easily be observed as the time between twoconsecutive vehicles crossing the stop line. In this case,however, the observations and analyses are equivalent to theempirical regression technique .

Assuming undersaturated conditions, however, the time each ofthe minor street vehicles spends in the first position could bemeasured as well. Again, the inverse of the average of thesetimes is the capacity. Examples of measured results are given byKyte et al. (1991).

From a theoretical point of view, this method is correct. Theproblems relate to the measurement techniques (e.g. by videotaping). Here it is quite difficult to define the beginning and theend of the time spent in the first position in a consistent way. Ifthis problem is solved, this method provides an easy procedurefor estimating the capacity for a movement from the minor streeteven if this traffic stream is not operating at capacity.

Following a study of AWSC intersections, Kyte et al. (1996)developed empirical equations for the departure headways froman approach for different levels of conflict.

h = h + h P + h P + h P (8.122)i b-i LT-adj LT RT-adj RT HV-adj HVwhere:

h is the adjusted saturation headway for thei

degree of conflict case i;h is the base saturation headway for case i;b-i

hLT-adj

and h are the headway adjustment factors for leftRT-adjand right turners respectively;

P and P are the proportion of left and right turners;LT RT

h is the adjustment factor for heavy vehicles; andHV-adj

P is the proportion of heavy vehicles.HV

The average departure headway, , is first assumed to be fourseconds and the degree of saturation, x , is the product of theflow rate, V and . A second iterative value of is given bythe equation:

where P(C ) is the probability that conflict C occurs. Thesei i

values also depend on estimates of and the h values. Theiservice time is given by the departure headway minus the move-up time.

Kyte et al. (1996) recognizes that capacity can be evaluated fromtwo points of view. First, the capacity can be estimatedassuming all other flows remain the same. This is the approachthat is typically used in Section 8.4.1. Alternatively capacity canbe estimated assuming the ratio of flow rates for differentmovements for all approaches remain constant. All flows areincrementally increased until one approach has a degree ofsaturation equal to one.

The further evaluation of these measurement results correspondsto the methods of the empirical regression techniques. Again,regression techniques can be employed to relate the capacityestimates to the traffic volumes in those movements with ahigher rank of priority.

8.11 Conclusions

This chapter describes the theory of unsignalized intersections has been extended to predict delays in the simplerwhich probably have the most complicated intersection control conditions.mechanism. The approaches used to evaluate unsignalizedintersections fall into three classes. (b) Queuing theory in which the service time attributes are

(a) Gap acceptance theory which assumes a mechanism for driver departure patterns. The advantages of usingdrivers departure. This is generally achieved with the queuing theory is that measures of delay (and queuenotion of a critical gap and a follow on time. This lengths) can be more easily defined for some of the moreattributes of the conflicting stream and the non priority complicated cases.stream are also required. This approach has beensuccessfully used to predict capacity (Kyte et al. 1996) and

described. This is a more abstract method of describing


8 - 43

(c) Simulation programs. These are now becoming more Research in these three approaches will undoubtably continue.popular. However, as a word of caution, the output from New theoretical work highlights parameters or issues that shouldthese models is only as good as the algorithms used in be considered further. At times, there will be a number ofthe model, and some simpler models can give excellent counter balancing effects which will only be identified throughresults. Other times, there is a temptation to look at the theory.output from models without relating the results to theexisting theory. This chapter describes most of the The issues that are likely to be debated in the future include thetheories for unsignalised intersections and should assist extent that one stream affects another as discussed in Sectionsimulation modelers to indicate useful extension to 8.6; the similarities between signalized and unsignalizedtheory. intersections; performance of oversaturated intersection and

variance associated with the performance measures.

References

Abou-Henaidy, M., S. Teply, and J. D. Hunt (1994). Gap Brilon, W. (1995). Delays at Oversaturated UnsignalizedAcceptance Investigations in Canada. Proceedings of the Intersections based on Reserve Capacities. Paper presentedSecond International Symposium on Highway Capacity. at the TRB Annual Meeting, Washington D.C.Australian Road Research Board- Transportation ResearchBoard, Vol 1. pp. 1-20.

Adams (1936). Road Traffic Considered as a Random Series.Journal of the Institute of Civil Engineers, London.Vol. 4, pp. 121-130.

Akcelik, R. and R. J. Troutbeck (1991). Implementation of theAustralian Roundabout Analysis Method in SIDRA. In:U. Brannolte (ed.) Highway Capacity and Level of Service, Washington D.C.Proceedings of the International Symposium on Highway Borel, G. (1942). Sur L'emploi du Théorème De BernoulliCapacity, Karlsruhe, A. A. Balkema, Rotterdam, pp.17-34. Pour Faciliter Le Calcul D'un Infinité De Co-Efficients.

Ashworth, R. (1968). A Note on the Selection of Gap Application Au Problème De L'attente À Un Guichet.Acceptance Criteria for Traffic Simulation Studies.Transportation Research, 2, pp. 171- 175, 1968. Buckley, D. J. (1962). Road Traffic Headway Distribution.

Boehm, H. (1968). Die Anwendung der Monte-Carlo-Methode in der Strassenverkehrstechnik. Teil I:Untersuchungen an ungesteuerten Knoten (Application ofthe Monte- Carlo-Method in Traffic Engineering. Part I:Investigations at Unsignalized Intersections).Schriftenreihe Strassenbau und Strassenverkehrstechnik,Vol. 73. Bonn 1968.

Brilon, W. (1988). Recent Developments in CalculationMethods for Unsignalized Intersections in West Germany.In: Intersections Without Traffic Signals (Ed.: W. Brilon).Springer publications, Berlin 1988.

Brilon, W. (Ed.) (1991). Intersections Without Traffic SignalsII. Springer Publications, Berlin.

Brilon, W. and M. Grossmann (1991). The New GermanGuideline for Capacity of Unsignalized Intersections. In:Intersections without Traffic Signals II (Ed.: W. Brilon).Springer Publications, Berlin.

Brilon, W, N. Wu, and K. Lemke (1996). Capacity atUnsignalized Two-Stage Priority Intersections. Paper961280. Presented at the 75th TRB Annual Meeting,

C. R. Acad. Sci. Paris Vol. 214, pp. 452-456.

Proceedings, 1st ARRB Conf., Vol. 1(1), pp. 153-186.Buckley, D. J. (1968). A Semi-Poisson Model of Traffic Flow.

Transportation Science, Vol. 2(2), pp. 107-132.Catchpole, E. A. and A. W. Plank (1986). The Capacity of a

Priority Intersection. Transportation Research Board, 20B(6), pp. 441-456

Chan, P. L. and S. Teply, S. (1991). Simulation of Multilane Stop-Controlled T-Intersections by Knosimo in Canada. In:Intersections without Traffic Signals II (Ed.: W. Brilon).Springer Publications, Berlin.

Cowan, R. J. (1975). Useful Headway Models.Transportation Research, 9(6), pp. 371-375.

Cowan, R. J. (1987). An Extension of Tanner's Results onUncontrolled Intersections. Queuing Systems, Vol 1., pp.249-263.


8 - 44

Daganzo, C. F. (1977). Traffic Delay at Unsignalized Hewitt, R. H. (1985). A Comparison Between Some MethodsIntersections: Clarification of Some Issues. Transportation of Measuring Critical Gap. Traffic Engineering & Control,Science, Vol. 11. 26(1), pp. 13-21.

Dawson, R. F. (1969). The Hyperlang Probability Heidemann, D. (1991). Queue Lengths and Waiting-Time Distribution - A Generalized Traffic Headway Model. Distributions at Priority Intersections. TransportationProceedings of the FourthISTTT in Karsruhe, Strassenbau Research B, Vol 25B, (4) pp. 163-174.und Strassenverkehehrstechnik, No 89 1969, pp 30-36.Drew, D. R. (1968). Traffic Flow Theory and Control.McGraw-Hill Book Company, New York.

FGSV (1991). Merkblatt zur Berechnung derLeistungsfähigkeit von Knotenpunkten ohneLichtsignalanlagen (Guideline for the Estimation ofCapacity at Unsignalized Intersections). German Vorlesungsmanuskript, Stuttgart.Association for Roads and Traffic (FGSV), Cologne.

Fisk, C. S. (1989). Priority Intersection Capacity - AGeneralization of Tanner's Formula. Transportation LR909.Research, Vol. 23 B, pp. 281-286.

Gipps, P. G. (1982). Some Modified Gap AcceptanceFormulae. Proceedings of the 11th ARRB Conference, Vol. SR 582.11, part 4, p.71.

Grossmann, M. (1988). Knosimo - a Practicable SimulationModel for Unsignalized Intersections. In: Intersectionswithout Traffic Signals (Ed.: W. Brilon). SpringerPublications, Berlin.

Grossmann, M. (1991). Methoden zur Berechnung undBeurteilung von Leistungsfähigkeit und Verkehrsqualitat anKnotenpunkten ohne Lichtsignalanlagen (Methods forCalculation and Judgement of Capacity and Traffic Qualityat Intersections Without Traffic Signals). Ruhr-UniversityBochum, Germany, Chair of Traffic Engineering, Vol. 9,.

Haight, F. A. and M. A. Breuer, (1960). The Borel-TannerDistribution. Biometrika, Vol. 47 (1 and 2), pp. 143-150.

Harders, J. (1968). Die Leistungsfähigkeit NichtSignalgeregelter Städtischer Verkehrsknoten (The Capacityof Unsignalized Urban Intersections). SchriftenreiheStrassenbau und Strassenverkehrstechnik, Vol. 76.

Harders, J. (1976). Grenz- und Folgezeitlücken als Grundlagefür die Leistungsfähigkeit von Landstrassen (Critical Gapsand Move-Up Times as the Basis of Capacity Calculations Cooperative Highway Research Program Project 3-46.for Rural Roads). Schriftenreihe Strassenbau und Little, J. (1961). A Proof of the Queueing Formula L = W.Strassenverkehrstechnik, Vol. 216. Operations Research 9, pp. 383-387.

HCM (1985). Highway Capacity Manual. Transportation McDonald, M. and D. J. Armitage (1978). The Capacity ofResearch Board, Special Report 209, Washington, DC.

Herbert, J. (1963). A Study of Four Way Stop Intersection pp. 447-450.Capacities. Highway Research Record, 27, HighwayResearch Board, Washington, DC.

Hewitt, R. H. (1983). Measuring Critical Gap.Transportation Science, 17(1), pp. 87-109.

Horowitz, A. J. (1993). Revised Queueing Model of Delay atAll-Way Stop-Controlled Intersections. TransportationResearch Record, 1398, pp. 49-53.

Jacobs, F. (1979). Leistungsfähigkeitsberechnung vonKnotenpunkten ohne Lichtsignalanlagen (CapacityCalculations for Unsignalised Intersections).

Kimber, R. M. and E. M. Hollis (1979). Traffic Queues andDelays at Road Junctions. TRRL Laboratory Report,

Kimber, R. M. and R. D. Coombe (1980). The TrafficCapacity of Major/Minor Priority Junctions. TRRL Report

Kremser, H. (1962). Ein Zusammengesetztes Wartezeitproblem Bei Poissonschen Verkehrsströmen (AComplex Problem of Delay with Poisson-Distributed TrafficFlows). Osterreichisches Ingenieur-Archiv, 16.

Kremser, H. (1964). Wartezeiten Und Warteschlangen BeiEinfadelung Eines Poissonprozesses in Einen AnderenSolchen Prozess (Delays and Queues with One PoissonProcess Merging into Another One). ÖsterreichischesIngenieur-Archiv, 18.

Kyte, M. (1989). Estimating Capacity and Delay at an All-Way Stop-Controlled Intersection. Research Report,TRANSNOW, Moscow/Idaho.

Kyte, M., J. Zegeer, and K. Lall (1991). Empirical Models forEstimating Capacity and Delay at Stop-ControlledIntersections in the United States. In: Intersections WithoutTraffic Signals II (Ed.: W. Brilon), Springer Publications,Berlin.

Kyte et al. (1996). Capacity and Level of Service atUnsignalized Intersections. Final Report for National

Roundabouts. Traffic Engineering & Control. Vol. 19(10),


8 - 45

Miller, A. J. (1972). Nine Estimators of Gap AcceptanceParameters. In: Traffic Flow and Transportation (Ed.Newell). Proceedings International Symposium on theTheory of Traffic Flow and Transportation, American Tanner, J. C. (1967). The Capacity of an UncontrolledElsevier Publishing Co.

Morse, P. M. (1962). Queues, Inventories and Maintenance.John Wiley.

Newell, G. F. (1971). Applications of Queueing Theory.Chapman and Hall Ltd., London.

Newell, G. F. (1982). Applications of Queueing Theory. 2ndEd. Chapman and Hall Ltd., London.

Plank, A. W. and E. A. Catchpole (1984). A General CapacityFormula for an Uncontrolled Intersection. TrafficEngineering Control. 25(6), pp. 327-329.

Poeschl, F. J. (1983). Die Nicht SignalgesteuerteNebenstrassenzufahrt Als Verallgemeinertes M/G/ 1-Warteschlangensystem (The Unsignalized Minor StreetEntry as a Generalized M/G/1 Queueing System). Zeitschriftfür Operations Research, Vol. 27 B.

Ramsey, J. B. H. and I. W. Routledge (1973). A NewApproach to the Analysis of Gap Acceptance Times. TrafficEngineering Control, 15(7), pp. 353-357.

Richardson, A. J. (1987). A Delay Model for Multiway Stop- Transportation and Traffic Theory, in Yokohama, Japan (Ed:Sign Intersections. Transportation Research Record, 1112,pp. 107-112. Troutbeck, R. J. (1991). Unsignalized Intersection and

Salter, R. J. (1982). Simulation of Delays and Queue Lengths Roundabouts in Australia: Recent Developments. In:at Over-Saturated Priority Highway Junctions. AustralianRoad Research, Vol. 12. No. 4.

Schuhl, A. (1955). The Probability Theory Applied to theDistribution of Vehicles on Two-Lane Highways. Poissonand Traffic. The Eno Foundation for Highway Traffic Arterial Roads. Physical Infrastructure Centre Report, 92-Control. Sangatuck, CT. 21, Queensland University of Technology, Brisbane.

Siegloch, W. (1973). Die Leistungsermittlung an Wegmann, H. (1991). A General Capacity Formula forKnotenpunkten Ohne Lichtsignalsteuerung (Capacity Unsignalized Intersections. In: Intersections without TrafficCalculations for Unsignalized Intersections). SchriftenreiheStrassenbau und Strassenverkehrstechnik, Vol. 154.

Steierwald, G. (1961). Die Anwendung Der Monte-Carlo-Methode in Der Strassenverkehrsforschung (TheApplication of the Monte-Carlo Method in TrafficEngineering Research). Habilitation thesis, RWTH Aachen.

Steierwald, G. (1961). Die Leistunsfähigkeit vonKnotenpurlkten des Strassenverkehrs. SchriftenreiheStrassenbau und Strassenverkehrstechnik, Vol. 11, Bonn.

Tanner, J. C. (1953). A Problem of Interference Between TwoQueues. Biometrika, 40, pp. 58-69.

Tanner, J. C. (1961) A Derivation of the Borel Distribution.Biometrica, Vol. 48, p. 222.

Tanner, J. C. (1962). A Theoretical Analysis of Delays At AnUncontrolled Intersection. Biometrica 49(1 and 2),pp. 163-70.

Intersection. Biometrica, 54(3 and 4), pp. 657-658.Tracz, M. (1991). Polish Guidelines for Capacity Analysis of

Priority Intersections. In: Intersections Without TrafficSignals II (Ed.: W. Brilon), Springer Publications, Berlin.

Troutbeck, R. J. (1975). A Review of the Ramsey-RoutledgeMethod for Gap Acceptance Times. Traffic Engineering &Control, 16(9), pp. 373-375.

Troutbeck. R. J. (1986). A verage Delay at an UnsignalizedIntersection with Two Major Streams Each Having aDichotomized Headway Distribution. TransportationScience, 20(4), pp. 272-286.

Troutbeck, R. J. (1988). Current and Future AustralianPractices for the Design of Unsignalized Intersections. In:Intersections without Traffic Signals (Ed.: W. Brilon),Springer Publications, Berlin.

Troutbeck, R. J. (1990). Roundabout Capacity and Associated Delay. In: Transportation and Traffic Theory.Proceedings of the Eleventh International Symposium on

M Koshi), Elsevier.

Intersections without Traffic Signals II (Ed.: W. Brilon),Springer Publications, Berlin.

Sullivan, D. and R. J. Troutbeck (1993). RelationshipBetween the Proportion of Free Vehicles and Flow Rate on

Signals II (Ed.: W. Brilon), Springer Publications, Berlin1991.

Wu, N. (1994). An Approximation for the Distribution ofQueue Lengths at Signalized Intersections. Proceedings ofthe Second International Symposium on Highway Capacity.Australian Road Research Board - Transportation ResearchBoard, Vol 2., pp. 717-736.

Yeo, G. F. (1962). Single-Server Queues with ModifiedService Mechanisms. Journal Australia MathematicsSociety, Vol 2, pp. 499-502.

Yeo, G. F. and B. Weesakul (1964). Delays to Road Trafficat an Intersection. Journal of Applied Probability,pp. 297-310.


8 - 46

Zhang, X. (1988). The Influence of Partial Constraint on Chodur, J. and S. Gaca (1988). Simulation Studies of theDelay at Priority Junctions. In: Intersections without Effects of Some Geometrical and Traffic Factors on theTraffic Signals (Ed.: W. Brilon), Springer Publications,Berlin.

Additional References(Not cited in the Chapter 8)

Anveden, J. (1988). Swedish Research on UnsignalizedIntersections. In: Intersections Without Traffic Signals (Ed.:W. Brilon). Springer Publications, Berlin.

Ashworth, R. (1969). The Capacity of Priority-TypeIntersections with a Non-Uniform Distribution of CriticalAcceptance Gaps. Transportation Research, Vol. 3.

Ashworth, R. (1970). The Analysis and Interpretation of GapAcceptance Data. Transportation Research. No. 4,pp. 270-280.

Baas, K. G. (1987). The Potential Capacity of UnsignalizedIntersections. ITE-Journal, pp. 43-56.

Bang, K. L., A. Hansson, and B. Peterson (1978). SwedishCapacity Manual. Transportation Research Record 667, pp1-28.

Brilon, W. (1981). Anmerkungen Zur Leistungsermittlung vonKnotenpunkten Ohne Lichtsignalanlagen. (Some Notes onCapacity Calculations for Unsignalized Intersections).Strassenverkehrstechnik, Heft 1, pp. 20-25.

Brilon, W. (Ed.) (1988). Intersections Without Traffic Signals.Springer Publications, Berlin.

Brilon, W. and M. Grossmann (1989). Entwicklung EinesSimulationsmodells Für Knotenpunkte OhneLichtsignalanlagen (Development of a Simulation Modelfor Intersections Without Traffic Signals). SchriftenreiheStrassenbau und Strassenverkehrstechnik, Vol. 554, Bonn.

Brilon, W. and M. Grossmann (1991). AktualisiertesBerechnungsverfahren Für Knotenpunkte OhneLichtsignalanlagen (Actualized Calculation Procedure forIntersections Without Traffic Signals). SchriftenreiheStrassenbau und Strassenverkehrstechnik, Vol. 596, Bonn.

Brilon, W., M. Grossmann, and B. Stuwe (1991). Towards aNew German Guideline for Capacity of UnsignalizedIntersections. Transportation Research Record, No. 1320,Washington, DC.

Catling, I. (1977). A Time Dependent Approach to JunctionDelays. Traffic Engineering & Control, Vol. 18(11).Nov. 77, pp. 520-523, 536.

Capacity of Priority Intersections. In: Intersections WithoutTraffic Signals (Ed.: W. Brilon). Springer Publications,Berlin.

Cowan, R. J. (1984). Adams' Formula Revised. TrafficEngineering & Control, Vol. 25(5), pp 272- 274.

Fisk C. S. and H. H. Tan (1989). Delay Analysis for PriorityIntersections. Transportation Research, Vol. 23 B, pp 452-469.

Fricker, J. D., M. Gutierrez, and D. Moffett (1991). GapAcceptance and Wait Time at Unsignalized Intersections.In: Intersections without Traffic Signals II (Ed.: W. Brilon).Springer Publications, Berlin.

Hansson, A. (1978). Swedish Capacity Manual. StatensVägverk (National Road Administration) Intern Rapport NR24.

Hansson, A. (1987). Berakningsmetoder För Olika EffektmåttI Korningar, Del Iii: Korsningar Utan Trafiksignaler.(Procedures for Estimating Performance Measures ofIntersections. Part III: Intersections Without SignalControl.) Statens Vägverk (National Road Administration).

Hansson, A. and T. Bergh (1988). A New Swedish CapacityManual/CAPCAL 2. Proceedings 14th Australian RoadResearch Board Conference, Vol. 14(2), pp. 38-47.

Hawkes, A. G. (1965). Queueing for Gaps in Traffic.Biometricia 52 (1 and 2). pp. 79-85.

Hondermarcq, H. (1968). Essais De Priorité Á Gancle. 9thIntentional Study Week on Road Traffic Flow and Safety.Munich.

Japan Society of Traffic Engineers (1988). The Planning andDesign of At-Grade Intersections.

Jessen, G. D. (1968). Ein Richtlinienvorschlag Für DieBehandlung Der Leistungsfähigkeit von KnotenpunktenOhne Signalregelung (A Guideline Suggested for CapacityCalculations for Unsignalized Intersections).Strassenverkehrstechnik, No. 7/8.

Jirava, P. and P. Karlicky (1988). Research on UnsignalizedIntersections with Impact on the Czechoslovak DesignStandard. In: Intersections without Traffic Signals (Ed.: W.Brilon). Springer Publications, Berlin.

Kimber, R. M., M. Marlow, and E. W. Hollis (1977). Flow/Delay Relationships for Major/Minor PriorityJunctions. Traffic Engineering & Control, 18(11),pp. 516-519.


8 - 47

Kimber, R. M., I. Summersgill, and I. J. Burrow (1986). Tracz, M., J. Chodur, and St. Gondek (1990). The Use ofDelay Processes at Unsignalized Junctions: The Reserve Capacity and Delay as the ComplementaryInterrelation Between Queueing and Geometric and Measures of Junction Effectiveness. Proceedings of theQueueing Delay. Transportation Research Board, 20B(6),pp. 457-476.

Lassarre, S., P. Lejeune, and J. C. Decret (1991). GapAcceptance and Risk Analysis at Unsignalized Intersections. In: Intersections Without Traffic Signals II, Springer-Verlag, Berlin.

Middelham, F. (1986). Manual for the Use of FLEXSYT-IWith FLEXCOL-76. Ministerie van Verkeer en Waterstaat,The Hague. Troutbeck, R. J. (1989). Evaluating the Performance of a

Semmens, M. C. (1985). PICADY 2: an Enhanced Program Roundabout. Australian Road Research Board Specialto Model Capacities, Queues and Delays at Major/MinorPriority Junctions. TRRL Report RR36.

Siegloch, W. (1974). Ein Richtlinienvorschlag ZurLeistungsermittlung an Knotenpunkten OhneLichtsignalsteuerung (Capacity Calculations forUnsignalized Intersections). Strassenverkehrstechnik,Vol. 1.

Tracz, M. (1988). Research of Traffic Performance ofMajor/Minor Priority Intersections. In: IntersectionsWithout Traffic Signals (Ed.: W. Brilon), SpringerPublications, Berlin.

International Symposium 'Transportation and TrafficTheory", Yokohama.

Tracz, M. and J. Chodur (1991). Comparative Analysis ofMajor/Minor Priority Intersection Capacity Methods. In:Highway Capacity and Level of Service, A. A. Balkema,Rotterdam.

TRB (1991). Interim Material on Unsignalized IntersectionCapacity. Transportation Research Circular No. 373.

Report, 45.Troutbeck, R. J. (1990). Traffic Interactions at Roundabouts.

Proceedings of the 15th Australian Road Research BoardConference, 15(5), pp. 17-42.

Troutbeck, R. J. (1992). Estimating the Critical AcceptanceGap from Traffic Movements. Research Report, 92-5.

Turner, D. J., R. Singh, and Y. H. Cheong (1984). TheDevelopment of Empirical Equations for Capacity Analysisat Priority Junctions in Singapore. Singapore Transport,Vol. 3, No. 4, pp. 17-21.

Vasarhely, B. (1976). Stochastic Simulation of the Traffic ofan Uncontrolled Road Intersection. TransportationResearch, Vol. 10.

Professor, Civil Engineering Department, North Carolina State University, Box 7908, Raleigh, NC 15

276-7908

Assistant Professor, Purdue University, West LaFayette, IN 4790716

Principal, TransSmart Technologies, Inc., Madison, WI 5370517

TRAFFIC FLOW ATSIGNALIZED INTERSECTIONS

BY NAGUI ROUPHAIL15

ANDRZEJ TARKO16

JING LI17

I �

variance of the number of arrivals per cyclemean number of arrivals per cycle

B �

variance of number of departures during cyclemean number of departures during cycle


I = cumulative lost time for phase i (sec)iL = total lost time in cycle (sec)q = A(t) = cumulative number of arrivals from beginning of cycle starts until t,B = index of dispersion for the departure process,

c = cycle length (sec)C = capacity rate (veh/sec, or veh/cycle, or veh/h)d = average delay (sec)d = average uniform delay (sec)1d = average overflow delay (sec)2D(t) = number of departures after the cycle starts until time t (veh)e = green extension time beyond the time to clear a queue (sec)gg = effective green time (sec)G = displayed green time (sec)h = time headway (sec)i = index of dispersion for the arrival processq = arrival flow rate (veh/sec)Q = expected overflow queue length (veh)0Q(t) = queue length at time t (veh)r = effective red time (sec)R = displayed red time (sec)S = departure (saturation) flow rate from queue during effective green (veh/sec)t = timeT = duration of analysis period in time dependent delay modelsU = actuated controller unit extension time (sec)Var(.) = variance of (.)W = total waiting time of all vehicles during some period of time iix = degree of saturation, x = (q/S) / (g/c), or x = q/Cy = flow ratio, y = q/SY = yellow (or clearance) time (sec)� = minimum headway

9 - 1

9.TRAFFIC FLOW AT SIGNALIZED INTERSECTIONS

9.1 Introduction

The theory of traffic signals focuses on the estimation of delays have received greater attention since the pioneering work byand queue lengths that result from the adoption of a signal Webster (1958) and have been incorporated in manycontrol strategy at individual intersections, as well as on a intersection control and analysis tools throughout the world. sequence of intersections. Traffic delays and queues areprincipal performance measures that enter into the This chapter traces the evolution of delay and queue lengthdetermination of intersection level of service (LOS), in the models for traffic signals. Chronologically speaking, earlyevaluation of the adequacy of lane lengths, and in the estimation modeling efforts in this area focused on the adaptation of steady-of fuel consumption and emissions. The following material state queuing theory to estimate the random component of delaysemphasizes the theory of descriptive models of traffic flow, asopposed to prescriptive (i.e. signal timing) models. Therationale for concentrating on descriptive models is that a betterunderstanding of the interaction between demand (i.e. arrivalpattern) and supply (i.e. signal indications and types) at trafficsignals is a prerequisite to the formulation of optimal signalcontrol strategies. Performance estimation is based onassumptions regarding the characterization of the traffic arrivaland service processes. In general, currently used delay modelsat intersections are described in terms of a deterministic andstochastic component to reflect both the fluid and randomproperties of traffic flow.

The deterministic component of traffic is founded on the fluidtheory of traffic in which demand and service are treated ascontinuous variables described by flow rates which vary over thetime and space domain. A complete treatment of the fluidtheory application to traffic signals has been presented inChapter 5 of the monograph.

The stochastic component of delays is founded on steady-statequeuing theory which defines the traffic arrival and service timedistributions. Appropriate queuing models are then used toexpress the resulting distribution of the performance measures.The theory of unsignalized intersections, discussed in Chapter 8of this monograph, is representative of a purely stochasticapproach to determining traffic performance.

Models which incorporate both deterministic (often calleduniform) and stochastic (random or overflow) components oftraffic performance are very appealing in the area of trafficsignals since they can be applied to a wide range of trafficintensities, as well as to various types of signal control. They areapproximations of the more theoretically rigorous models, inwhich delay terms that are numerically inconsequential to thefinal result have been dropped. Because of their simplicity, they

and queues at intersections. This approach was valid so long asthe average flow rate did not exceed the average capacity rate.In this case, stochastic equilibrium is achieved and expectationsof queues and delays are finite and therefore can be estimated bythe theory. Depending on the assumptions regarding thedistribution of traffic arrivals and departures, a plethora ofsteady-state queuing models were developed in the literature.These are described in Section 9.3 of this chapter.

As traffic flow rate approaches or exceeds the capacity rate, atleast for a finite period of time, the steady-state modelsassumptions are violated since a state of stochastic equilibriumcannot be achieved. In response to the need for improvedestimation of traffic performance in both under and oversaturatedconditions, and the lack of a theoretically rigorous approach tothe problem, other methods were pursued. A prime example isthe time-dependent approach originally conceived by Whiting(unpublished) and further developed by Kimber and Hollis(1979). The time-dependent approach has been adopted in manycapacity guides in the U.S., Europe and Australia. Because it iscurrently in wide use, it is discussed in some detail in Section 9.4of this chapter.

Another limitation of the steady-state queuing approach is theassumption of certain types of arrival processes (e.g Binomial,Poisson, Compound Poisson) at the signal. While valid in thecase of an isolated signal, this assumption does not reflect theimpact of adjacent signals and control which may alter thepattern and number of arrivals at a downstream signal. Thereforeperformance in a system of signals will differ considerably fromthat at an isolated signal. For example, signal coordination willtend to reduce delays and stops since the arrival process will bedifferent in the red and green portions of the phase. The benefitsof coordination are somewhat subdued due to the dispersion ofplatoons between signals. Further, critical signals in a systemcould have a metering effect on traffic which proceeds


9 - 2

downstream. This metering reflects the finite capacity of the without reference to their impact on signal performance. Thecritical intersection which tends to truncate the arrival manner in which these controls affect performance is quitedistribution at the next signal. Obviously, this phenomenon has diverse and therefore difficult to model in a generalizedprofound implications on signal performance as well, fashion. In this chapter, basic methodological approaches andparticularly if the critical signal is oversaturated. The impact of concepts are introduced and discussed in Section 9.6. Aupstream signals is treated in Section 9.5 of this chapter. complete survey of adaptive signal theory is beyond the scope of

With the proliferation of traffic-responsive signal controltechnology, a treatise on signal theory would not be complete

this document.

9.2 Basic Concepts of Delay Models at Isolated Signals

As stated earlier, delay models contain both deterministic and time is that portion of green where flows are sustained at thestochastic components of traffic performance. The deterministic saturation flow rate level. It is typically calculated at thecomponent is estimated according to the following assumptions: displayed green time minus an initial start-up lost time (2-3a) a zero initial queue at the start of the green phase, b) a seconds) plus an end gain during the clearance interval (2-4uniform arrival pattern at the arrival flow rate (q) throughout thecycle c) a uniform departure pattern at the saturation flow rate(S) while a queue is present, and at the arrival rate when thequeue vanishes, and d) arrivals do not exceed the signal capacity,defined as the product of the approach saturation flow rate (S)and its effective green to cycle ratio (g/c). The effective green

seconds depending on the length of the clearance phase).

A simple diagram describing the delay process in shown inFigure 9.1. The queue profile resulting from this application isshown in Figure 9.2. The area under the queue profilediagram represents the total (deterministic) cyclic delay. Several

Figure 9.1 Deterministic Component of Delay Models.


9 - 3

Figure 9.2Queuing Process During One Signal Cycle

(Adapted from McNeil 1968).

performance measures can be derive including the average delay Interestingly, at extremely congested conditions, the stochasticper vehicle (total delay divided by total cyclic arrivals) the queuing effect are minimal in comparison with the size ofnumber of vehicle stopped (Q ), the maximum number ofs

vehicles in the queue (Q ) , and the average queue lengthmax

(Q ). Performance models of this type are applicable to lowavgflow to capacity ratios (up to about 0.50), since the assumptionof zero initial and end queues is not violated in most cases.

As traffic intensity increases, however, there is a increasedlikelihood of “cycle failures”. That is, some cycles will begin toexperience an overflow queue of vehicles that could notdischarge from a previous cycle. This phenomenon occurs atrandom, depending on which cycle happens to experiencehigher-than-capacity flow rates. The presence of an initial queue(Q ) causes an additional delay which must be considered in theoestimation of traffic performance. Delay models based on queuetheory (e.g. M/D/n/FIFO) have been applied to account for thiseffect.

oversaturation queues. Therefore, a fluid theory approach maybe appropriate to use for highly oversaturated intersections.This leaves a gap in delay models that are applicable to therange of traffic flows that are numerically close to the signalcapacity. Considering that most real-world signals are timed tooperate within that domain, the value of time-dependent modelsare of particular relevance for this range of conditions.

In the case of vehicle actuated control, neither the cycle lengthnor green times are known in advance. Rather, the length of thegreen is determined partly by controller-coded parameters suchas minimum and maximum green times, and partly by the patternof traffic arrivals. In the simplest case of a basic actuatedcontroller, the green time is extended beyond its minimum solong as a) the time headway between vehicle arrivals does notexceed the controller�s unit extension (U), and b) the maximumgreen has not been reached. Actuated control models arediscussed further in Section 9.6.

d �

c�gc(1�q/S)

[Qo

q�

c�g�12

]

I � var(A)qh

W � W1 � W2,

W1 � �(c�g)

0[Q(0)�A(t)]dt

W2 � �c

(c�g)Q(t)dt

E(W1) � (c�g)Qo �12

q (c�g)2.


9 - 4

(9.1)

(9.10)

(9.3)

(9.4)

(9.5)

(9.6)

9.3 Steady-State Delay Models

9.3.1 Exact Expressions

This category of models attempts to characterize traffic delaysbased on statistical distributions of the arrival and departureprocesses. Because of the purely theoretical foundation of themodels, they require very strong assumptions to be consideredvalid. The following section describes how delays are estimatedfor this class of models, including the necessary datarequirements.

The expected delay at fixed-time signals was first derived byBeckman (1956) with the assumption of the binomial arrivalprocess and deterministic service:

where,

c = signal cycle,g = effective green signal time,q = traffic arrival flow rate,S = departure flow rate from queue during green,Q = expected overflow queue from previous cycles.o

The expected overflow queue used in the formula and therestrictive assumption of the binomial arrival process reduce thepractical usefulness of Equation 9.1. Little (1961) analyzed theexpected delay at or near traffic signals to a turning vehiclecrossing a Poisson traffic stream. The analysis, however, did notinclude the effect of turners on delay to other vehicles. Darroch(1964a) studied a single stream of vehicles arriving at afixed-time signal. The arrival process is the generalized Poissonprocess with the Index of Dispersion:

where,var(.)= variance of ( . )q = arrival flow rate,h = interval length,A = number of arrivals during interval h = qh.

The departure process is described by a flexible service mecha-nism and may include the effect of an opposing stream by defin-ing an additional queue length distribution caused by this factor.Although this approach leads to expressions for the expectedqueue length and expected delay, the resulting models arecomplex and they include elements requiring further modelingsuch as the overflow queue or the additional queue componentmentioned earlier. From this perspective, the formula is not ofpractical importance. McNeil (1968) derived a formula for theexpected signal delay with the assumption of a general arrivalprocess, and constant departure time. Following his work, weexpress the total vehicle delay during one signal cycle as a sumof two components

whereW = total delay experienced in the red phase and1W = total delay experienced in the green phase.2

and

where,Q(t) = vehicle queue at time t,A(t) = cumulative arrivals at t,

Taking expectations in Equation 9.4 it is found that:

Let us define a random variable Z as the total vehicle delay2experienced during green when the signal cycle is infinite. The

E(Z2) �(1�Iq/S�q/S) E[Q(t0)]

2S (1�q/S)2�

E[Q 2(t0)]2S (1�q/S)

.

E(W2) � E[Z2�Q(t�c�g)]�E[Z2�Q(t�c)]

E[W2] �(1�Ig/S�q/S) �E[Q(c�g)�Q(c)]�

2S (1�q/S)2�

E[Q 2(c�g)] � E[Q 2(c)]2S (1�q/S)

.

x �

q/Sg/c

< 1.

E[Q(c�g)�Q(c)] � E[A(c�g)] � q(c�g)

E[Q 2(c�g)�Q 2(c)] � 2E[A(c�g)]E[Q(0)] �E[A 2(c�g)]

� 2q(c�g)Qo�q 2 (c�g)2�

q (c�g)I

E(W2) �1

2S (1�q/S)2[(1�Iq/S�q/S)g (c�g)

� (1�q/S)(2q(c�g)Qo �

q 2 (c�g)2� q (c�g)I)]

E(W) � (c�g)c(1�q/S)

Q0

q�

c�g2

qc �

1S

1� I1�q/S

d� c�g2c(1�q/S)

[(c�g) � 2q

Qo �1S

(1� I1�q/S

)]

d �

(c�g)2c(1�q/S)

(c�g)� 2q

1� (1�q/S)(1�B 2)2S

Qo�

1S

(1� I�B 2q/S1�q/S

)


9 - 5

(9.7)

(9.8)

(9.9)

(9.10)

(9.11)

(9.12)

(9.13)

(9.14)

(9.15)

(9.16)

variable Z is considered as the total waiting time in a busy2

period for a queuing process Q(t) with compound Poissonarrivals of intensity q, constant service time 1/S and an initialsystem state Q(t=t ). McNeil showed that provided q/S<1:0

Now W can be expressed using the variable Z :2 2

and

The queue is in statistical equilibrium, only if the degree ofsaturation x is below 1,

For the above condition, the average number of arrivals per cyclecan discharge in a single green period. In this case E[ Q(0)] = E [ Q(c) ] and E [ Q (0) ] = E [Q (c) ]. Also Q (c-g) = 2 2

Q(0) + A(c), so that:

and

Equations 9.9, 9.11, and 9.12 yield:

and using Equations 9.3, 9.4 and 9.13, the following is obtained:

The average vehicle delay d is obtained by dividing E(W) by theaverage number of vehicles in the cycle (qc):

which is in essence the formula obtained by Darroch when thedeparture process is deterministic. For a binomial arrivalprocess I=1-q/S, and Equation 9.15 becomes identical to thatobtained by Beckmann (1956) for binomial arrivals. McNeiland Weiss (in Gazis 1974) considered the case of the compoundPoisson arrival process and general departure process obtainingthe following model:

An examination of the above equation indicates that in the caseof no overflow (Q = 0), and no randomness in the traffic processo

(I=0), the resultant delay becomes the uniform delay component.This component can be derived from a simple input-outputmodel of uniform arrivals throughout the cycle and departures asdescribed in Section 9.2. The more general case in Equation9.16 requires knowledge of the size of the average overflowqueue (or queue at the beginning of green), a major limitation onthe practical usefulness of the derived formulae, since these areusually unknown.

d �

c(1�g/c)2

2[1�(g/c)x]�

x 2

2q(1�x)�0.65( c

q 2)

13x 2�5(g/c)

Q(c) � Q(0) � A � C ��C

E(�C) � E(C�A),

Q(c) � [�C�E(�C)] � Q(0) � [C�A�E(C�A)]

E[Q(c)]2�2E[Q(c)]E(�C]�Var(�C) �E[Q(0)]2

�Var(C�A)


9 - 6

(9.17)

(9.18)

(9.19)

(9.20)

(9.21)

A substantial research effort followed to obtain a closed-form signal performance, since vehicles are served only during theanalytical estimate of the overflow queue. For example, Haight effective green, obviously at a higher rate than the capacity rate.(1959) specified the conditional probability of the overflow The third term, calibrated based on simulation experiments, is aqueue at the end of the cycle when the queue at the beginning of corrective term to the estimate, typically in the range of 10the cycle is known, assuming a homogeneous Poisson arrival percent of the first two terms in Equation 9.17. process at fixed traffic signals. The obtained results were thenmodified to the case of semi-actuated signals. Shortly thereafter, Delays were also estimated indirectly, through the estimation ofNewell (1960) utilized a bulk service queuing model with anunderlying binomial arrival process and constant departure time,using generating function technique. Explicit expressions foroverflow queues were given for special cases of the signal split.

Other related work can be found in Darroch (1964a) who useda more general arrival distribution but did not produce a closedform expression of queue length, and Kleinecke (1964), whosework included a set of exact but complicated series expansionfor Q , for the case of constant service time and Poisson arrivaloprocess.

9.3.2 Approximate Expressions

The difficulty in obtaining exact expressions for delay which arereasonably simple and can cover a variety of real world condi-tions, gave impetus to a broad effort for signal delay estimationusing approximate models and bounds. The first, widely usedapproximate delay formula was developed by Webster (1961,reprint of 1958 work with minor amendments) from acombination of theoretical and numerical simulation approaches:

where,d = average delay per vehicle (sec),c = cycle length (sec),g = effective green time (sec),x = degree of saturation (flow to capacity ratio),q = arrival rate (veh/sec).

The first term in Equation 9.17 represents delay when traffic canbe considered arriving at a uniform rate, while the second termmakes some allowance for the random nature of the arrivals.This is known as the "random delay", assuming a Poisson arrivalprocess and departures at constant rate which corresponds to thesignal capacity. The latter assumption does not reflect actual

Q , the average overflow queue. Miller (1963) for example ob-o

tained a approximate formulae for Q that are applicable to anyoarrival and departure distributions. He started with the generalequality true for any general arrival and departure processes:

where,Q(c) = vehicle queue at the end of cycle,Q(0) = vehicle queue at the beginning of cycle,A = number of arrivals during cycle,C = maximum possible number of departures

during green,�C = reserve capacity in cycle equal to

(C-Q(0)-A) if Q(0)+A < C , zero otherwise.

Taking expectation of both sides of Equation 9.18, Millerobtained:

since in equilibrium Q(0) = Q(c).

Now Equation 9.18 can be rewritten as:

Squaring both sides, taking expectations, the following isobtained:

Qo�Var(C�A)�Var(�C)

2E(C�A)

Qo�Var(C�A)2E(C�A)

Qo�Ix

2(1�x)

I�Var(�C)E(C�A)

Qo�(2x�1)I2(1�x)

, x �0.50

d �

(1�g/c)2(1�q/s)

c(1�g/c) �2Q0

q

Qo�exp �1.33 Sg(1�x)/x

2(1�x).

d �

c(1�g/c)2

2(1�q/S)�

Qo

q.


9 - 7

(9.22)

(9.23)

(9.24)

(9.25)

(9.26)

(9.27)

(9.28)

(9.29)

For equilibrium conditions, Equation 9.21 can be rearranged as which can now be substituted in Equation 9.15. Furtherfollows: approximations of Equation 9.15 were aimed at simplifying it for

where,

C = maximum possible number of departures inone cycle,

A = number of arrivals in one cycle,�C = reserve capacity in one cycle.

The component Var(�C) is positive and approaches 0 whenE(C) approaches E(A). Thus an upper bound on the expectedoverflow queue is obtained by deleting that term. Thus:

For example, using Darroch's arrival process (i.e. E(A)=qc,Var(A)=Iqc) and constant departure time during green(E(C)=Sg, Var(C)=0) the upper bound is shown to be:

where x=(qc)/(Sg).

Miller also considered an approximation of the excluded termVar(�C). He postulates that:

and thus, an approximation of the overflow queue is

practical purposes by neglecting the third and fourth terms whichare typically of much lower order of magnitude than the first twoterms. This approach is exemplified by Miller (1968a) whoproposed the approximate formula:

which can be obtained by deleting the second and third terms inMcNeil's formula 9.15. Miller also gave an expression for theoverflow queue formula under Poisson arrivals and fixed servicetime during the green:

Equations 9.15, 9.16, 9.17, 9.27, and 9.28 are limited to specificarrival and departure processes. Newell (1965) aimed at devel-oping delay formulae for general arrival and departure distribu-tions. First, he concluded from a heuristic graphical argumentthat for most reasonable arrival and departure processes, thetotal delay per cycle differs from that calculated with theassumption of uniform arrivals and fixed service times (Clayton,1941), by a negligible amount if the traffic intensity is sufficient-ly small. Then, by assuming a queue discipline LIFO (Last InFirst Out) which does not effect the average delay estimate, heconcluded that the expected delay when the traffic is sufficientlyheavy can be approximated:

This formula gives identical results to formula (Equation 9.15)if one neglects components of 1/S order in (Equation 9.15) andwhen 1-q/S=1-g/c. The last condition, however, is never met ifequilibrium conditions apply. To estimate the overflow queue,Newell (1965) defines F as the cumulative distribution of theQ

overflow queue length, F as the cumulative distribution of theA-D

overflow in the cycle, where the indices A and D representcumulative arrivals and departures, respectively. He showed thatunder equilibrium conditions:

FQ(x)��

0FQ(z)dFA�D(x�z)

Qo �qc(1�x)� �

�/2

0

tan2�

�1�exp[Sg(1�x)2/(2cos2�)]

d�.

Qo �

IH(μ)x2(1�x)

.

μ� Sg�qc(ISg)1/2

.

d �

c(1�g/c)2

2(1�q/S)�

Qo

q�

(1�g/c)I2S(1�q/S)2

.


9 - 8

(9.30)

(9.31)

(9.32)

(9.33)

(9.34)

The integral in Equation 9.30 can be solved only under therestrictive assumption that the overflow in a cycle is normallydistributed. The resultant Newell formula is as follows:

A more convenient expression has been proposed by Newell inthe form:

where,

The function H(μ) has been provided in a graphical form.

Moreover, Newell compared the results given by expressions(Equation 9.29) and (Equation 9.31) with Webster's formula andadded additional correction terms to improve the results formedium traffic intensity conditions. Newell's final formula is:

Table 9.1Maximum Relative Discrepancy between the Approximate Expressionsand Ohno's Algorithm (Ohno 1978).

Range of y = 0.0 �� 0.5 Range of g/c = 0.4 �� 1.0

Approximate Expressions s = 0.5 v/s s = 1.5 v/s s = 0.5 v/s s = 1.5 v/s (Equation #, Q computed according 0

to Equation #)

c = 90 s c = 30 s c = 90 s c = 30 s c = 30 s c = 90 s

g = 46 s g = 16 s g = 45.33 s g = 15.33 s q = 0.2 s q = 0.6 s

Modified Miller's expression (9.15, 0.22 2.60 -0.53 0.22 2.24 0.269.28)

Modified Newell's expression (9.15, 0.82 2.53 0.25 0.82 2.83 0.25-9.31)

McNeil's expression (9.15, Miller 1969) 0.49 1.79 0.12 0.49 1.51 0.08

Webster's full expression (9.17) -8.04 -21.47 3.49 -7.75 119.24 1381.10

Newell's expression (9.34, 9.31) -4.16 10.89 -1.45 -4.16 -15.37 -27.27

H(μ) � exp[�μ�(μ2/2)]

μ�(1�x) (Sg)1/2.

A(t)��t

0q(�)d�

D(t) � �t

0S(�)d�


9 - 9

(9.35)

(9.36)

(9.37)

(9.38)

More recently, Cronje (1983b) proposed an analytical overflow queue calculated with the method described byapproximation of the function H(μ):

where,

He also proposed that the correction (third) component in Equa-tion 9.34 could be neglected.

Earlier evaluations of delay models by Allsop (1972) andHutchinson (1972) were based on the Webster model form.Later on, Ohno (1978) carried out a comparison of the existing categorized by the g/c ratio. Further efforts to improve on theirdelay formulae for a Poisson arrival process and constant estimates will not give any appreciable reduction in the errors.departure time during green. He developed a computational The modified Miller expression was recommended by Ohnoprocedure to provide the basis for evaluating the selected because of its simpler form compared to McNeil's and Newell's.models, namely McNeil's expression, Equation 9.15 (with

Miller 1969), McNeil's formula with overflow queue accordingto Miller (Equation 9.28) (modified Miller's expression),McNeil's formula with overflow queue according to Newell(Equation 9.31) (modified Newell's expression), Websterexpression (Equation 9.17) and the original Newell expression(Equation 9.34). Comparative results are depicted in Table 9.1and Figures 9.3 and 9.4. Newell's expression appear to be moreaccurate than Webster, a conclusion shared by Hutchinson(1972) in his evaluation of three simplified models (Newell,Miller, and Webster). Figure 9.3 represents the percentagerelative errors of the approximate delay models measured againstOhno’s algorithm (Ohno 1978) for a range of flow ratios. Themodified Miller's and Newell's expressions give almost exactaverage delay values, but they are not superior to the originalMcNeil formula. Figure 9.4 shows the same type of errors,

9.4 Time-Dependent Delay Models

The stochastic equilibrium assumed in steady-state models parabolic, or triangular functions) and calculates the correspond-requires an infinite time period of stable traffic conditions ing delay. In May and Keller (1967) delay and queues are calcu-(arrival, service and control processes) to be achieved. At low lated for an unsignalized bottleneck. Their work is neverthelessflow to capacity ratios equilibrium is reached in a reasonable representative of the deterministic modeling approach and canperiod of time, thus the equilibrium models are an acceptable be easily modified for signalized intersections. The generalapproximation of the real-world process. When traffic flow ap- assumption in their research is that the random queueproaches signal capacity, the time to reach statistical equilibrium fluctuations can be neglected in delay calculations. The modelusually exceeds the period over which demand is sustained. defines a cumulative number of arrivals A(t):Further, in many cases the traffic flow exceeds capacity, asituation where steady-state models break down. Finally, trafficflows during the peak hours are seldom stationary, thus violatingan important assumption of steady-state models. There hasbeen many attempts at circumventing the limiting assumptionof steady-state conditions. The first and simplest way is to dealwith arrival and departure rates as a function of time in adeterministic fashion. Another view is to model traffic at signals,assuming stationary arrival and departure processes but notnecessarily under stochastic equilibrium conditions, in order toestimate the average delay and queues over the modeled periodof time. The latter approach approximates the time-dependentarrival profile by some mathematical function (step-function,

and departures D(t) under continuous presence of vehiclequeue over the period [0,t]:


9 - 10

Figure 9.3Percentage Relative Errors for Approximate Delay

Models by Flow Ratios (Ohno 1978).

Figure 9.4Relative Errors for Approximate Delay Models

by Green to Cycle Ratios (Ohno 1978).

Q(t)�Q(0)�A(t)�D(t)

d� 1A(T)�

T

0Q(t)dt

d�d1�T2

(x�1)

Q(T) � Q(0) � (x�1)CT .


9 - 11

(9.39)

(9.40)

(9.41)

(9.42)

The current number of vehicles in the system (queue) is

and the average delay of vehicles queuing during the time period[0,T] is

The above models have been applied by May and Keller to atrapezoidal-shaped arrival profile and constant departure rate.One can readily apply the above models to a signal with knownsignal states over the analysis period by substituting C(�) forS(�) in Equation 9.38:

C(�) = 0 if signal is red,= S(�) if signal is green and Q(�) > 0,= q(�) if signal is green and Q(�) = 0.

Deterministic models of a single term like Equation 9.39 yieldacceptable accuracy only when x<<1 or x>>1. Otherwise, they version of the Pollaczek-Khintchine equation (Taha 1982), hetend to underestimate queues and delays since the extra queues illustrated the calculation of average queue and delay for eachcauses by random fluctuations in q and C are neglected.

According to Catling (1977), the now popular coordinatetransformation technique was first proposed by Whiting, who didnot publish it. The technique when applied to a steady-statecurve derived from standard queuing theory, produces a time-dependent formula for delays. Delay estimates from the newmodels when flow approaches capacity are far more realisticthan those obtained from the steady-state model. The followingobservations led to the development of this technique.

� At low degree of saturation (x<<1) delay is almost equalto that occurring when the traffic intensity is uniform(constant over time).

� At high degrees of saturation (x>>1) delay can besatisfactory described by the following deterministic modelwith a reasonable degree of accuracy:

where d is the delay experienced at very low traffic1

intensity, (uniform delay) T = analysis period over whichflows are sustained.

� steady-state delay models are asymptotic to the y-axis (i.egenerate infinite delays) at unit traffic intensity (x=1). Thecoordinate transformation method shifts the originalsteady-state curve to become asymptotic to thedeterministic oversaturation delay line--i.e.-- the secondterm in Equation 9.41--see Figure 9.5. The horizontaldistance between the proposed delay curve and itsasymptote is the same as that between the steady-statecurve and the vertical line x=1.

There are two restrictions regarding the application of theformula: (1) no initial queue exists at the beginning of theinterval [0,T], (2) traffic intensity is constant over the interval[0,T]. The time-dependent model behaves reasonably within theperiod [0,T] as indicated from simulation experiments. Thus,this technique is very useful in practice. Its principal drawback,in addition to the above stated restrictions (1) and (2) is the lackof a theoretical foundation. Catling overcame the latter diffi-culties by approximating the actual traffic intensity profile witha step-function. Using an example of the time-dependent

time interval starting from an initial, non-zero queue.

Kimber and Hollis (1979) presented a computational algorithmto calculate the expected queue length for a system with randomarrivals, general service times and single channel service(M/G/1). The initial queue can be defined through itsdistribution. To speed up computation, the average initial queueis used unless it is substantially different from the queue atequilibrium. In this case, the full computational algorithmshould be applied. The non-stationary arrival process is approxi-mated with a step-function. The total delay in a time period iscalculated by integrating the queue size over time. Thecoordinate transformation method is described next in somedetail.

Suppose, at time T=0 there are Q(0) waiting vehicles in queueand that the degree of saturation changes rapidly to x. In a deter-ministic model the vehicle queue changes as follows:

Q� x �

Bx 2

1�x

B � 0.5 1 �

�2

μ2

1 � x � xd � xT

x � xT � (xd�1)


9 - 12

(9.43)

(9.44)(9.45)

(9.46)

Figure 9.5The Coordinate Transformation Method.

The steady-state expected queue length from the modified The following derivation considers the case of exponentialPollaczek-Khintczine formula is:

where B is a constant depending on the arrival and departureprocesses and is expressed by the following equation.

where � and μ are the variance and mean of the service time2

distribution, respectively.

service times, for which � = μ , B =1. Let x be the degree of2 2d

saturation in the deterministic model (Equation 9.42), x refers tothe steady-state conditions in model (Equation 9.44), while xT

refers to the time-dependent model such that Q(x,T)=Q(x ,T).TTo meet the postulate of equal distances between the curves andthe appropriate asymptotes, the following is true from Figure9.5:

and hence

xd �Q(T)�Q(0)

CT� 1,

x � xT �Q(T)�Q(0)

CT.

Q(T)1�Q(T)

� xT �

Q(T)�Q(0)CT

Q(T)� 12

[(a 2�b)1/2

�a]

a�(1�x)CT�1�Q(0)

b � 4[Q(0) � xCT].

a � (1�x)(CT)2�[1�Q(0)]CT�2(1�B)[Q(0)�xCT]

CT�(1�B)

b �

4[Q(0) � xCT][CT � (1�B)(Q(0) � xCT)]CT � (1�B)

.

dd�

[Q(0)�1]� 12

(x�1)CT

C

ds �1C

(1� Bx1�x

) .

d� 12

[(a 2�b)1/2

�a]

a� T2

(1�x)� 1C

[Q(0)�B�2]

b �

4C

[ T2

(1�x) � 12

xTB �

Q(0)�1C

(1�B)] .


9 - 13

(9.47)

(9.48)

(9.49)

(9.50)

(9.51)

(9.52)

(9.53)

(9.54)

(9.55)

(9.56)

(9.57)

(9.58)

(9.59)

and from Equation 9.42:

the transformation is equivalent to setting: The equation for the average delay for vehicles arriving during

From Figure 9.5, it is evident that the queue length at time T,Q(T) is the same at x, x , and x . By substituting for Q(T) inT dEquation 9.44, and rewriting Equation 9.48 gives: and the steady-state delay d ,

By eliminating the index T in x and solving the second degreeT

polynomial in Equation 9.49 for Q(T), it can be shown that:

where

and

If the more general steady state Equation 9.43 is used, the resultfor Equation 9.51 and 9.52 is:

and

the period of analysis is also derived starting from the averagedelay per arriving vehicle d over the period [0,T],d

s

The transformed time dependent equation is

with the corresponding parameters:

and

The derivation of the coordinate transformation technique hasbeen presented. The steady-state formula (Equation 9.43) doesnot appear to adequately reflect traffic signal performance, sincea) the first term (queue for uniform traffic) needs furtherelaboration and b) the constant B must be calibrated for casesthat do not exactly fit the assumptions of the theoretical queuingmodels.

Qo �

1.5(x�xo)1�x

when x>xo,

0 otherwise

xo � 0.67 � Sg600

Qo �

CT4

[(x�1)� (x�1)2�

12(x�xo)CT

] when x>xo,

0 otherwise.

d �

c(1�g/c)2

2(1�q/S)when x<1

(c�g)/2 when x�1

�

Qo

C.

Pi,j(t)��

C�0Pi,j(t,C)P(C)

Pi,0(t,C) ��C�i

k�0P(t,A�k) when i�C,

0 otherwise

Pi, j(t,C)�P(t,A� j�i�C) when j� i�C,

0 otherwise .

PQ(t) � PQ(t�1) P(t) .


9 - 14

(9.60)

(9.61)

(9.62)

(9.63)

(9.64)

(9.65)

(9.66)

(9.67)

Akçelik (1980) utilized the coordinate transformation techniqueto obtain a time-dependent formula which is intended to be moreapplicable to signalized intersection performance than Kimber-Hollis's. In order to facilitate the derivation of a time-dependentfunction for the average overflow queue Q , Akçelik used the Cronje (1983a), and Miller (1968a); Olszewski (1990) usedofollowing expression for undersaturated signals as a simple non-homogeneous Markov chain techniques to calculate theapproximation to Miller's second formula for steady-state queuelength (Equation 9.28):

where

Akçelik's time-dependent function for the average overflowqueue is

The formula for the average uniform delay during the interval[0,T] for vehicles which arrive in that interval is

Generalizations of Equations 9.60 and 9.61 were discussed byAkçelik (1988) and Akçelik and Rouphail (1994). It should benoted that the average overflow queue, Q is an approximation0of the McNeil (Equation 9.15) and Miller (Equation 9.28)formulae applied to the time-dependent conditions, and differsfrom Newell's approximations Equation 9.29 and Equation 9.34of the steady-state conditions. According to Akçelik (1980), this

approximation is relevant to high degrees of saturation x and itseffect is negligible for most practical purposes.

Following certain aspects of earlier works by Haight (1963),

stochastic queue distribution using the arrival distribution P(t,A)and capacity distribution P(C). Probabilities of transition froma queue of i to j vehicles during one cycle are expressed by thefollowing equation:

and

and

The probabilities of queue states transitions at time t form thetransition matrix P(t). The system state at time t is defined withthe overflow queue distribution in the form of a row vector P (t).Q

The initial system state variable distribution at time t =0 isassumed to be known: P (0)=[P (0), P (0),...P (0)], where P (0)Q 1 2 m i

is the probability of queue of length i at time zero. The vector ofstate probabilities in any cycle t can now be found by matrixmultiplication:

Equation 9.67, when applied sequentially, allows for the calcula-tion of queue probability evolution from any initial state.

In their recent work, Brilon and Wu (1990) used a similarcomputational technique to Olszewski's (1990a) in order to


9 - 15

evaluate existing time-dependent formulae by Catling (1977), which incorporate the impact of the arrival profile shape (e.g. theKimber-Hollis (1979), and Akçelik (1980). A comparison of peaking intensity) on delay. In this examination of delay modelsthe models results is given in Figures 9.6 and 9.7 for a parabolic in the time dependent mode, delay is defined according to thearrival rate profile in the analysis period T . They found that theoCatling method gives the best approximation of the averagedelay. The underestimation of delays observed in the Akçelik's vehicle, even if this time occurs beyond the analysis period T.model is interpreted as a consequence of the authors' using an The path trace method will tend to generate delays that areaverage arrival rate over the analyzed time period instead of the typically longer than the queue sampling method, in whichstep function, as in the Catling's method. When the peak flow stopped vehicles are sampled every 15-20 seconds for therate derived from a step function approximation of the parabolic duration of the analysis period. In oversaturated conditions, theprofile is used in Akçelik's formula, the results were virtually measurement of delay may yield vastly different results asindistinguishable from Brilon and Wu's (Akçelik and Rouphail vehicles may discharge 15 or 30 minutes beyond the analysis1993). period. Thus it is important to maintain consistency between

Using numeric results obtained from the Markov Chains discussion of the delay measurement methods and their impactapproach, Brilon and Wu developed analytical approximate (and on oversaturation delay estimation, the reader is referred torather complicated) delay formulae of a form similar to Akçelik's Rouphail and Akçelik (1992a).

Figure 9.6Comparison of Delay Models Evaluated by Brilonand Wu (1990) with Moderate Peaking (z=0.50).

path trace method of measurement (Rouphail and Akçelik1992a). This method keeps track of the departure time of each

delay measurements and estimation methods. For a detailed

Figure 9.7Comparison of Delay Models Evaluated by Brilon

and Wu (1990) with High Peaking (z=0.70).

f(�) � D�

2� 2�

exp �( D�

�

D�

)2

2�2

q2(t2)dt2 � �t1q1(t1) f(t2�t1) dt1 dt2

q2(j) � �i q1(i)g(j�i)

q2(j) �1

1�a�q1(j) � (1� 1

1�a�) q2(j�1)

�


9 - 16

(9.68)

(9.69)

(9.70)

(9.71)

9.5 Effect of Upstream Signals

The arrival process observed at a point located downstream ofsome traffic signal is expected to differ from that observedupstream of the same signal. Two principal observations aremade: a) vehicles pass the signal in "bunches" that are separatedby a time equivalent to the red signal (platooning effect), and b)the number of vehicles passing the signal during one cycle doesnot exceed some maximum value corresponding to the signalthroughput (filtering effect).

9.5.1 Platooning Effect On Signal Performance

The effect of vehicle bunching weakens as the platoon movesdownstream, since vehicles in it travel at various speeds,spreading over the downstream road section. This phenomenon,known as platoon diffusion or dispersion, was modeled by Pacey(1956). He derived the travel time distribution f(�) along a roadsection assuming normally distributed speeds and unrestrictedovertaking:

where,

D = distance from the signal to the point where arrivalsare observed,

� = individual vehicle travel time along distance D,= mean travel time, and

� = standard deviation of speed.

The travel time distribution is then used to transform a trafficflow profile along the road section of distance D:

where,

q (t )dt = total number of vehicles passing some2 2 2point downstream of the signal in theinterval (t, t+dt),

q (t )dt = total number of vehicles passing the1 1 1

signal in the interval (t, t+dt), andf(t -t ) = probability density of travel time (t - t )2 1 2 1

according to Equation 9.68.

The discrete version of the diffusion model in Equation 9.69 is

where i and j are discrete intervals of the arrival histograms.

Platoon diffusion effects were observed by Hillier and Rothery(1967) at several consecutive points located downstream ofsignals (Figure 9.8). They analyzed vehicle delays at pretimedsignals using the observed traffic profiles and drew the followingconclusions:

� the deterministic delay (first term in approximate delayformulae) strongly depends on the time lag between thestart of the upstream and downstream green signals(offset effect);

� the minimum delay, observed at the optimal offset,increases substantially as the distance between signalsincreases; and

� the signal offset does not appear to influence theoverflow delay component.

The TRANSYT model (Robertson 1969) is a well-knownexample of a platoon diffusion model used in the estimation ofdeterministic delays in a signalized network. It incorporates theRobertson's diffusion model, similar to the discrete version of thePacey's model in Equation 9.70, but derived with the assumptionof the binomial distribution of vehicle travel time:

where � is the average travel time and a is a parameter whichmust be calibrated from field observations. The Robertsonmodel of dispersion gives results which are satisfactory for the


9 - 17

Figure 9.8Observations of Platoon Diffusion

by Hillier and Rothery (1967).

purpose of signal optimization and traffic performance analysis In the TRANSYT model, a flow histogram of traffic servedin signalized networks. The main advantage of this model over (departure profile) at the stopline of the upstream signal is fir stthe former one is much lower computational demand which is a constructed, then transformed between two signals using modelcritical issue in the traffic control optimization for a large size (Equation 9.71) in order to obtain the arrival patterns at thenetwork. stopline of the downstream signal. Deterministic delays at

the downstream signal are computed using the transformedarrival and output histograms.

fp �PVGg/c

B � I exp(�1.3F 0.627)

F �

Qo

Ia qc


9 - 18

(9.72)

(9.73)

(9.74)

To incorporate the upstream signal effect on vehicle delays, the The remainder of this section briefly summarizes recent workHighway Capacity Manual (TRB 1985) uses a progression factor pertaining to the filtering effect of upstream signals, and the(PF) applied to the delay computed assuming an isolated signal.A PF is selected out of the several values based on a platoonratio f . The platoon ratio is estimated from field measurementpand by applying the following formula:

where,

PVG = percentage of vehicles arriving during theeffective green,

g = effective green time,c = cycle length.

Courage et al. (1988) compared progression factor valuesobtained from Highway Capacity Manual (HCM) with thoseestimated based on the results given by the TRANSYT model.They indicated general agreement between the methods,although the HCM method is less precise (Figure 9.9). To avoidfield measurements for selecting a progression factor, theysuggested to compute the platoon ratio f from the ratios ofpbandwidths measured in the time-space diagram. They showedthat the proposed method gives values of the progression factorcomparable to the original method.

Rouphail (1989) developed a set of analytical models for directestimation of the progression factor based on a time-spacediagram and traffic flow rates. His method can be considered asimplified version of TRANSYT, where the arrival histogramconsists of two uniform rates with in-platoon and out-of-platoontraffic intensities. In his method, platoon dispersion is also basedon a simplified TRANSYT-like model. The model is thussensitive to both the size and flow rate of platoons. Morerecently, empirical work by Fambro et al. (1991) and theoreticalanalyses by Olszewski (1990b) have independently confirmedthe fact that signal progression does not influence overflowqueues and delays. This finding is also reflected in the mostrecent update of the Signalized Intersections chapter of theHighway Capacity Manual (1994). More recently, Akçelik(1995a) applied the HCM progression factor concept to queuelength, queue clearance time, and proportion queued at signals.

resultant overflow delays and queues that can be anticipated atdownstream traffic signals.

9.5.2 Filtering Effect on Signal Performance

The most general steady-state delay models have been derivedby Darroch (1964a), Newell (1965), and McNeil (1968) for thebinomial and compound Poisson arrival processes. Since theseefforts did not deal directly with upstream signals effect, thequestion arises whether they are appropriate for estimatingoverflow delays in such conditions. Van As (1991) addressedthis problem using the Markov chain technique to model delaysand arrivals at two closely spaced signals. He concluded that theMiller's model (Equation 9.27) improves random delay estima-tion in comparison to the Webster model (Equation 9.17).Further, he developed an approximate formula to transform thedispersion index of arrivals, I , at some traffic signal into thedispersion index of departures, B, from that signal:

with the factor F given by

This model (Equation 9.73) can be used for closely spacedsignals, if one assumes the same value of the ratio I along a roadsection between signals.

Tarko et al. (1993) investigated the impact of an upstream signalon random delay using cycle-by-cycle macrosimulation. Theyfound that in some cases the ratio I does not properly representthe non-Poisson arrival process, generally resulting in delayoverestimation (Figure 9.10).

They proposed to replace the dispersion index I with anadjustment factor f which is a function of the difference betweenthe maximum possible number of arrivals m observable duringc

one cycle, and signal capacity Sg:


9 - 19

Figure 9.9HCM Progression Adjustment Factor vs Platoon Ratio

Derived from TRANSYT-7F (Courage et al. 1988).

f � 1�e a(mc�Sg)


9 - 20

(9.75)

Figure 9.10Analysis of Random Delay with Respect to the Differential Capacity Factor (f)

and Var/Mean Ratio of Arrivals (I)- Steady State Queuing Conditions (Tarko et al. 1993).

where a is a model parameter, a < 0.

A recent paper by Newell (1990) proposes an interestinghypothesis. The author questions the validity of using randomdelay expressions derived for isolated intersections at internalsignals in an arterial system. He goes on to suggest that the sumof random delays at all intersections in an arterial system with noturning movements is equivalent to the random delay at thecritical intersection, assuming that it is isolated. Tarko et al.(1993) tested the Newell hypothesis using a computational

model which considers a bulk service queuing model and a setof arrival distribution transformations. They concluded thatNewell's model estimates provide a close upper bound to theresults from their model. The review of traffic delay models atfixed-timed traffic signals indicate that the state of the art hasshifted over time from a purely theoretical approach groundedin queuing theory, to heuristic models that have deterministicand stochastic components in a time-dependent domain. Thismove was motivated by the need to incorporate additional factorssuch as non-stationarity of traffic demand, oversaturation, trafficplatooning and filtering effect of upstream signals. It isanticipated that further work in that direction will continue,with a view towards using the performance-based models forsignal design and route planning purposes.

9.6 Theory of Actuated and Adaptive Signals

The material presented in previous sections assumed fixed time traffic-adaptive systems requires new delay formulations that aresignal control, i.e. a fixed signal capacity. The introduction of sensitive to this process. In this section, delay models fortraffic-responsive control, either in the form of actuated or actuated signal control are presented in some detail, which

E{W1j}�q1

2(1�q1/S1)(E{rj}�Y)2

�Var(rj)

�

I1(E{rj}�Y)S1(1�q1/S1)

�

V1

S1q1

E{W2j}�q2

2(1�q2/S2)[(E{gj}�Y)2

�Var(gj)�I2[E(gj)�Y]S2(1�q2/S2)

�

V2

S2q2


9 - 21

(9.76)

(9.77)

incorporate controller settings such as minimum and maximum and maximum greens, the phase will be extended for eachgreens and unit extensions. A brief discussion of the state of the arriving vehicle, as long as its headway does not exceed theart in adaptive signal control follows, but no models are value of unit extension. An intersection with two one-waypresented. For additional details on this topic, the reader is streets was studied. It was found that, associated with eachencouraged to consult the references listed at the end of the traffic flow condition, there is an optimal vehicle interval forchapter. which the average delay per vehicle is minimized. The value of

9.6.1 Theoretically-Based Expressions

As stated by Newell (1989), the theory on vehicle actuatedsignals and related work on queues with alternating priorities isvery large, however, little of it has direct practical value. Forexample, "exact" models of queuing theory are too idealized tobe very realistic. In fact the issue of performance modeling ofvehicle actuated signals is too complex to be described by acomprehensive theory which is simple enough to be useful.Actuated controllers are normally categorized into: fully-actuated, semi-actuated, and volume-density control. To date,the majority of the theoretical work related to vehicle actuatedsignals is limited to fully and semi-actuated controllers, but notto the more sophisticated volume-density controllers withfeatures such as variable initial and extension intervals. Twotypes of detectors are used in practice: passage and presence.Passage detectors, also called point or small-area detectors,include a small loop and detect motion or passage when avehicle crosses the detector zone. Presence detectors, also calledarea detectors, have a larger loop and detect presence of vehiclesin the detector zone. This discussion focuses on traffic actuatedintersection analysis with passage detectors only.

Delays at traffic actuated control intersections largely depend onthe controller setting parameters, which include the followingaspects: unit extension, minimum green, and maximum green.Unit extension (also called vehicle interval, vehicle extension, orgap time) is the extension green time for each vehicle as itarrives at the detector. Minimum green: summation of the initialinterval and one unit extension. The initial interval is designedto clear vehicles between the detector and the stop line.Maximum green: the maximum green times allowed to a specificphase, beyond which, even if there are continuous calls for thecurrent phase, green will be switched to the competing approach.

The relationship between delay and controller setting parametersfor a simple vehicle actuated type was originally studied byMorris and Pak-Poy (1967). In this type of control, minimumand maximum greens are preset. Within the range of minimum

the optimal vehicle interval decreases and becomes more critical,as the traffic flow increases. It was also found that by using theconstraints of minimum and maximum greens, the efficiency andcapacity of the signal are decreased. Darroch (1964b) alsoinvestigated a method to obtain optimal estimates of the unitextension which minimizes total vehicle delays.

The behavior of vehicle-actuated signals at the intersection oftwo one-way streets was investigated by Newell (1969). Thearrival process was assumed to be stationary with a flow rate justslightly below the saturation rate, i.e. any probabilitydistributions associated with the arrival pattern are timeinvariant. It is also assumed that the system is undersaturatedbut that traffic flows are sufficiently heavy, so that the queuelengths are considerably larger than one car. No turningmovements were considered. The minimum green isdisregarded since the study focused on moderate heavy trafficand the maximum green is assumed to be arbitrarily large. Nospecific arrival process is assumed, except that it is stationary.

Figure 9.11 shows the evolution of the queue length when thequeues are large. Traffic arrives at a rate of q , on one approach,1

and q , on the other. r , g , and Y represent the effective red,2 j j j

green, and yellow times in cycle j. Here the signal timings arerandom variables, which may vary from cycle to cycle. For anyspecific cycle j, the total delay of all cars W is the area of aijtriangular shaped curve and can be approximated by:

Var(rj)�Var(r), Var(gj)�Var(g)

E{Wkj}�E{Wk}, k�1,2

E{rj}�E{r}, E{gj}�E{g}

E{r}�Yq2/S2

1�q1/S1�q2/S2

E{g}�Yq1/S1

1�q1/S1�q2/S2


9 - 22

(9.79)

(9.80)

(9.78)

(9.81)

(9.82)

Figure 9.11Queue Development Over Time Under

Fully-Actuated Intersection Control (Newell 1969).

where

E{W }, E{W } = the total wait of all cars during1j 2j

cycle j for approach 1 and 2;S , S = saturation flow rate for approach 11 2

and 2;E{r }, E{g } = expectation of the effective redj j

and green times;Var(r ), Var(g ) = variance of the effective red andj j

green time;I , I = variance to mean ratio of arrivals1 2

for approach 1 and 2; andV , V = the constant part of the variance of1 2

departures for approach 1 and 2.

Since the arrival process is assumed to be stationary,

The first moments of r and g were also derived based on theproperties of the Markov process:

Variances of r and g were also derived, they are not listed herefor the sake of brevity. Extensions to the multiple lane casewere investigated by Newell and Osuna (1969).

A delay model with vehicle actuated control was derived byDunne (1967) by assuming that the arrival process follows abinomial distribution. The departure rates were assumed to beconstant and the control strategy was to switch the signal whenthe queue vanishes. A single intersection with two one-lane one-way streets controlled by a two phase signal was considered.

For each of the intervals (k�, k�+�), k=0,1,2... the probability ofone arrival in approach i = 1, 2 is denoted by q and thei

probability of no arrival by p =1-q . The time interval, �, isi itaken as the time between vehicle departures. Saturation flowrate was assumed to be equal for both approaches. Denote

W (2)r�1�W (2)

r �μ[�1�c��2]

μ�0 with probability p2 ,1 with probability q2 .

E(W (2)r�1)�E(W (2)

r )�q2(r�1)/p2

E(W (2)r )�q2(r

2�r)/2p2

E(W (2))�q2{var[r]E 2[r]�E[r]}/(2p2)

F(t)� 1��e ��(t��) for t��0 for t<�

E(g1)�q1L

1�q1�q2

E(g2)�q2L

1�q1�q2

E(r1)�l2�q2L

1�q1�g2

E(r2)�l1�q1L

1�q1�q2


9 - 23

(9.83)

(9.84)

(9.85)

(9.86)

(9.87)

(9.88)

(9.89)

(9.90)

(9.91)

(9.92)

W as the total delay for approach 2 for a cycle having effective (2)r

red time of length r, then it can be shown that:

where c is the cycle length, � , and � are increases in delay at1 2the beginning and at the end of the cycle, respectively, when onevehicle arrives in the extra time unit at the beginning of thephase and:

Equation 9.83 means that if there is no arrivals in the extra timeunit at the beginning of the phase, then W =W , otherwise (2) (2)

r+1 rW =W + � + c + � . (2) (2)

r+1 r 1 2

Taking the expectation of Equation 9.83 and substituting forE(� ), E(� ):1 2

Solving the above difference equation for the initial condition W=0 gives,(2)

0

Finally, taking the expectation of Equation 9.86 with respect tor gives

Therefore, if the mean and variance of (r) are known, delay canbe obtained from the above formula. E(W ) for approach 2 is (i)

obtained by interchanging the subscripts.

Cowan (1978) studied an intersection with two single-lane one-way approaches controlled by a two-phase signal. The controlpolicy is that the green is switched to the other approach at theearliest time, t, such that there is no departures in the interval[t-� -1, t]. In general � � 0. It was assumed that departurei iheadways are 1 time unit, thus the arrival headways are at least1 time unit. The arrival process on approach j is assumed tofollow a bunched exponential distribution. It comprises random-

sized bunches separated by inter-bunch headways. All bunchedvehicles are assumed to have the same headway of 1 time unit.All inter-bunch headways follow the exponential distribution.Bunch size was assumed to have a general probabilitydistribution with mean, μ , and variance, � . The cumulativej j

2

probability distribution of a headway less than t seconds, F(t), is

where,� = minimum headway in the arrival stream, �=1

time unit;� = proportion of free (unbunched) vehicles; and� = a delay parameter.

Formulae for average signal timings (r and g) and average delaysfor the cases of � = 0 and � > 0 are derived separately. � = 0j j jmeans that the green ends as soon as the queues for the approachclear while � > 0 means that after queues clear there will be ajpost green time assigned to the approach. By analyzing theproperty of Markov process, the following formula are derivedfor the case of � = 0.j

L(1�q2)2(1�q1�q2)

�

q 21 (1�q2)

2�2(�

22�μ2

2)�(1�q1)3(1�q2)�1(�

21�μ2

1)2(1�q1�q2)(1�q1�g2�2q1q2)

g�gmax � ge<gemax

g � gs � eg

gmin�g�gmax

gs�fq yr1�y

eg � nghg � et

eg� �

1q� (�

��

1q

)e q(U��)

d � 0.9(d1�d2) � 0.9[ c(1�g/c)2

2(1�q/s)�

x 2

2q(1�x)]


9 - 24

(9.93)

(9.94)

(9.95)

(9.96)

(9.97)

(9.98)

(9.99)

(9.100)

where, The saturated portion of green period can be estimated from the

E(g ), E(g ) = expected effective green for1 2approach 1 and 2;

E(r ), E(r ) = expected effective red for approach 11 2and 2;

L = lost time in cycle;l , l = lost time for phase 1 and 2; and1 2q , q = the stationary flow rate for1 2

approach 1 and 2.

The average delay for approach 1 is:

Akçelik (1994, 1995b) developed an analytical method forestimating average green times and cycle time at a basic vehicleactuated controller that uses a fixed unit extension setting byassuming that the arrival headway follows the bunched where,exponential distribution proposed by Cowan (1978). In hismodel, the minimum headway in the arrival stream � is notequal to one. The delay parameter, �, is taken as �q /�, wheret

q is the total arrival flow rate and �=1-�q . In the model, the change after queue clearance; andt tfree (unbunched) vehicles are defined as those with headwaysgreater than the minimum headway �. Further, all bunchedvehicles are assumed to have the same headway �. Akçelik(1994) proposed two different models to estimate the proportionof free (unbunched) vehicles �. The total time, g, allocated to amovement can be estimated as where g is the minimum greenmin

time and g , the green extension time. This green time, g, isesubject to the following constraint

where g and g are maximum green and extension timemax emaxsettings separately. If it is assumed that the unit extension is setso that a gap change does not occur during the saturated portionof green period, the green time can be estimated by:

where g is the saturated portion of the green period and e is thes gextension time assuming that gap change occurs after the queueclearance period. This green time is subject to the boundaries:

following formula:

where,

f = queue length calibration factor to allow forqvariations in queue clearance time;

S = saturation flow;r = red time; andy = q/S, ratio of arrival to saturation flow rate.

The average extension time beyond the saturated portion can beestimated from:

n = average number of arrivals before a gapgchange after queue clearance;

h = average headway of arrivals before a gapg

e = terminating time at gap change (in most caset

it is equal to the unit extension U).

For the case when e = U, Equation 9.98 becomest

9.6.2 Approximate Delay Expressions

Courage and Papapanou (1977) refined Webster's (1958) delaymodel for pretimed control to estimate delay at vehicle-actuatedsignals. For clarity, Webster's simplified delay formula isrestated below.

Courage and Papapanou used two control strategies: (1) theavailable green time is distributed in proportion to demand on

c0�1.5L�51��yci

ca�1.5L

1��yci

d � d1� DF � d2

d1�c(1�g/c)2

2(1�xg/c)

d2�900Tx 2[(x�1)� (x�1)2�

mxCT

]

ca�Lxc

xc��yci

gi�yci

xcca


9 - 25

(9.101)

(9.102)

(9.103)

(9.104)

(9.105)

(9.106)

(9.107)

the critical approaches; and (2) wasted time is minimized byterminating each green interval as the queue has been properlyserviced. They propose the use of the cycle lengths shown inTable 9.2 for delay estimation under pretimed and actuatedsignal control:

Table 9.2 Cycle Length Used For Delay Estimation for Fixed-Time and Actuated Signals Using Webster’sFormula (Courage and Papapanou 1977).

Type of Signal in 1st Term in 2nd TermCycle Length Cycle Length

Pretimed Optimum Optimum The delay factor DF=0.85, reduces the queuing delay to account

Actuated Average Maximum

The optimal cycle length, c , is Webster's:0

where L is total cycle lost time and y is the volume to saturationci

flow ratio of critical movement i. The average cycle length, c isadefined as:

and the maximum cycle length, c , is the controller maximummaxcycle setting. Note that the optimal cycle length under pretimedcontrol will generally be longer than that under actuated control.The model was tested by simulation and satisfactory resultsobtained for a wide range of operations.

In the U. S. Highway Capacity Manual (1994), the averageapproach delay per vehicle is estimated for fully-actuatedsignalized lane groups according to the following:

where, d, d , d , g, and c are as defined earlier and1 2

DF = delay factor to account for signal coordination andcontroller type;

x = q/C, ratio of arrival flow rate to capacity;m = calibration parameter which depends on the arrival

pattern;C = capacity in veh/hr; andT = flow period in hours (T=0.25 in 1994 HCM).

for the more efficient operation with fully-actuated operationwhen compared to isolated, pretimed control. In an upcomingrevision to the signalized intersection chapter in the HCM, thedelay factor will continue to be applied to the uniform delay termonly.

As delay estimation requires knowledge of signal timings in theaverage cycle, the HCM provides a simplified estimationmethod. The average signal cycle length is computed from:

where x = critical q/C ratio under fully-actuated control (x =0.95c c

in HCM). For the critical lane group i, the effective green:

This signal timing parameter estimation method has been thesubject of criticism in the literature. Lin (1989), among others,compared the predicted cycle length from Equation 9.106 withfield observations in New York state. In all cases, the observedcycle lengths were higher than predicted, while the observed xcratios were lower.

Lin and Mazdeysa(1983) proposed a general delay model of thefollowing form consistent with Webster's approximate delayformula:

d�0.9�

c(1�K1gc

)2

2(1�K1gc

K2 x)�

3600(K2x)2

2q(1�K2x)�

tn� nw�

2(nL�si)a

� Gmini

eni�stn

s�q

ei � ��

n�nmin

Pj(n/Ti)eni

1�pj(n<nmin)

J �

�Ui�

tf(t)dtF(h<Ui)

Ei��

k�0(kJ�U)[F(h�U)]kF(h>U)�� 1q�[��1

q]e q(U��)


9 - 26

(9.10)

(9.109)

(9.110)

(9.111)

(9.112)

(9.113)

where g, c, q, x are as defined earlier and K and K are two1 2coefficients of sensitivity which reflect different sensitivities oftraffic actuated and pretimed delay to both g/c and x ratios. Inthis study, K and K are calibrated from the simulation model1 2for semi-actuated and fully-actuated control separately. Moreimportantly, the above delay model has to be used in conjunctionwith the method for estimating effective green and cycle length.In earlier work, Lin (1982a, 1982b) described a model toestimate the average green duration for a two phase fully-actuated signal control. The model formulation is based on thefollowing assumption: (1) the detector in use is small areapassage detector; (2) right-turn-on-red is either prohibited or itseffect can be ignored; and (3) left turns are made only fromexclusive left turn lanes. The arrival pattern for each lane wasassumed to follow a Poisson distribution. Thus, the headwaydistribution follows a shifted negative exponential distribution.

Figure 9.12 shows the timing sequence for a two phase fullyactuated controller. For phase i, beyond the initial greeninterval, g , green extends for F based on the control logic andmini i

the settings of the control parameters. F can be further dividedi

into two components: (1) e — the additional green extended bynin vehicles that form moving queues upstream of the detectorsafter the initial interval G ; (2) E — the additional greenmini ni

extended by n vehicles with headways of no more than one unitextension, U, after G or e . Note that e and E are randommini ni ni nivariables that vary from cycle to cycle. Lin (1982a, 1982b)developed the procedures to estimate e and E , the expectedi i

value of e and E , as follows. A moving queue upstream of ani ni

detector may exist when G is timed out in case the flow ratemini

of the critical lane q is high. If there are n vehicles arriving inc

the critical lane during time T , then the time required for the nthi

vehicle to reach the detector after G is timed out can beminiestimated by the following equation:

where w is the average time required for each queuing vehicle tostart moving after the green phase starts, L is the averagevehicle length, a is the vehicle acceleration rate from a standingposition., and s is the detector setback. If t �0, there is non

moving queue exists and thus e =0; otherwise the green will bei

extended by the moving queue. Let s be the rate atwhich the queuing vehicle move across the detector.Considering that additional vehicles may join the queue duringthe time interval t , if t >0 and s>0, then:n n

To account for the probability that no moving queues existupstream of the detector at the end of the initial interval, theexpected value of e , e is expressed as:ni i

where n is the minimum number of vehicles required to formmina moving queue.

To estimate E , let us suppose that after the initial G andi mini

additional green e have elapsed, there is a sequence of kni

consecutive headways that are shorter than U followed by aheadway longer than U. In this case the green will be extendedk times and the resultant green extension time is kJ+U withprobability [F(h � U)] F(h � U), where J is the average lengthk

of each extension and F(h) is the cumulative headwaydistribution function.

and therefore

where � is the minimum headway in the traffic stream.

Gi � ��

n�0(Gmini�ei�Ei)P(n/Ti)

Gmini�ei�Ei � (Gmax)i d2�900T[(x�1)� (x�1)2�

8kxCT

]

d2�kx

C(1�x)


9 - 27

(9.114)

(9.115) (9.116)

(9.117)

Figure 9.12Example of a Fully-Actuated Two-Phase Timing Sequence (Lin 1982a).

Referring to Figure 9.12, after the values of T and T are1 2

obtained, G can be estimated as: i

subject to

where P(n/T ) is the probability of n arrivals in the critical lanei

of the ith phase during time interval T . Since both T and T arei 1 2

unknown, an iterative procedure was used to determine G and1G .2

Li et al. (1994) proposed an approach for estimating overflowdelays for a simple intersection with fully-actuated signal

control. The proposed approach uses the delay format in the1994 HCM (Equations 9.104 and 9.105) with some variations,namely a) the delay factor, DF, is taken out of the formulationof delay model and b) the multiplier x is omitted from the2

formulation of the overflow delay term to ensure convergence tothe deterministic oversaturated delay model. Thus, the overflowdelay term is expressed as:

where the parameter (k) is derived from a numerical calibrationof the steady-state for of Equation 9.105 as shown below.


9 - 28

This expression is based on a more general formula by Akçelik models would satisfy the requirement that both controls yield(1988) and discussed by Akçelik and Rouphail (1994). The identical performance under very light and very heavy trafficcalibration results for the parameter k along with the overallstatistical model evaluation criteria (standard error and R ) are2

depicted in Table 9.3. The parameter k which corresponds topretimed control, calibrated by Tarko (1993) is also listed. It isnoted that the pretimed steady-state model was also calibratedusing the same approach, but with fixed signal settings. The firstand most obvious observation is that the pretimed modelproduced the highest k (delay) value compared to the actuatedmodels. Secondly, the parameter was found to increase with thesize of the controller's unit extension (U).

Procedures for estimating the average cycle length and greenintervals for semi-actuated signal operations have beendeveloped by Lin (1982b, 1990) and Akçelik (1993b). Recently,Lin (1992) proposed a model for estimating average cycle lengthand green intervals under semi-actuated signal control operationswith exclusive pedestrian-actuated phase. Luh (1991) studiedthe probability distribution of and delay estimation for semi-actuated signal controllers.

In summary, delay models for vehicle-actuated controllers arederived from assumptions related to the traffic arrival process,and are constrained by the actuated controller parameters. Thedistribution of vehicle headways directly impact the amount ofgreen time allocated to an actuated phase, while controllerparameters bound the green times within specified minimumsand maximums. In contrast to fixed-time models, performancemodels for actuated have the additional requirement ofestimating the expected signal phase lengths. Further researchis needed to incorporate additional aspects of actuated operationssuch as phase skipping, gap reduction and variable maximumgreens. Further, there is a need to develop generalized modelsthat are applicable to both fixed time and actuated control. Such

demands. Recent work along these lines has been reported byAkçelik and Chung (1994, 1995).

9.6.3 Adaptive Signal Control

Only a very brief discussion of the topic is presented here.Adaptive signal control systems are generally consideredsuperior to actuated control because of their true demandresponsiveness. With recent advances in microprocessortechnology, the gap-based strategies discussed in the previoussection are becoming increasingly outmoded and demonstrablyinefficient. In the past decade, control algorithms that rely onexplicit intersection/network delay minimization in a time-variant environment, have emerged and been successfully tested.While the algorithms have matured both in Europe and the U.S.,evident by the development of the MOVA controller in the U.K.(Vincent et al. 1988), PRODYN in France (Henry et al. 1983),and OPAC in the U.S. (Gartner et al. 1982-1983), theoreticalwork on traffic performance estimation under adaptive controlis somewhat limited. An example of such efforts is the work byBrookes and Bell (1991), who investigated the use of MarkovChains and three heuristic approaches in an attempt to calculatethe expected delays and stops for discrete time adaptive signalcontrol. Delays are computed by tracing the queue evolutionprocess over time using a `rolling horizon' approach. The mainproblem lies in the estimation (or prediction) of the initial queuein the current interval. While the Markov Chain approach yieldstheoretically correct answers, it is of limited value in practice dueto its extensive computational and storage requirements.Heuristics that were investigated include the use of the meanqueue length, in the last interval as the starting queue in thisinterval; the `two-spike' approach, in which the queue length

Table 9.3Calibration Results of the Steady-State Overflow Delay Parameter (k) (Li et al. 1994).

Control Pretimed U=2.5 U=3.5 U=4.0 U=5.0k (m=8k) 0.427 0.084 0.119 0.125 0.231s.e. NA 0.003 0.002 0.002 0.006R2 0.903 0.834 0.909 0.993 0.861


9 - 29

distribution has non-zero probabilities at zero and at an integer Overall, the latter method was recommended because it not onlyvalue closest to the mean; and finally a technique that propagates produces estimates that are sufficiently close to the theoreticalthe first and second moment of the queue length distribution estimate, but more importantly it is independent of the trafficfrom period to period. arrival distribution.

9.7 Concluding Remarks

In this chapter, a summary and evolution of traffic theory arrival process at the intersection, and of traffic meteringpertaining to the performance of intersections controlled by which may causes a truncation in the departure distribution fromtraffic signals has been presented. The focus of the discussion a highly saturated intersection. Next, an overview of delaywas on the development of stochastic delay models. models which are applicable to intersections operating under

Early models focused on the performance of a single intersectionexperiencing random arrivals and deterministic service timesemulating fixed-time control. The thrust of these models hasbeen to produce point estimates--i.e. expectations of-- delay andqueue length that can be used for timing design and quality ofservice evaluation. The model form typically include adeterministic component to account for the red-time delay and astochastic component to account for queue delays. The latterterm is derived from a queue theory approach.

While theoretically appealing, the steady-state queue theoryapproach breaks down at high degrees of saturation. Theproblem lies in the steady-state assumption of sustained arrivalflows needed to reach stochastic equilibrium (i.e the probabilityof observing a queue length of size Q is time-independent) . In further attention and require additional research. To begin with,reality, flows are seldom sustained for long periods of time and the assumption of uncorrelated arrivals found in most models istherefore, stochastic equilibrium is not achieved in the field at not appropriate to describe platooned flow--where arrivals arehigh degrees of saturation. highly correlated. Secondly, the estimation of the initial

A compromise approach, using the coordinate transformation and documented. There is also a need to develop queuing/delaymethod was presented which overcomes some of these models that are constrained by the physical space available fordifficulties. While not theoretically rigorous, it provides a means queuing. Michalopoulos (1988) presented such an applicationfor traffic performance estimation across all degrees of saturation using a continuous flow model approach. Finally, models thatwhich is also dependent on the time interval in which arrival describe the interaction between downstream queue lengths andflows are sustained. upstream departures are needed. Initial efforts in this direction

Further extensions of the models were presented to take into Rouphail and Akçelik (1992b). account the impact of platooning, which obviously alter the

vehicle actuated control was presented. They include stochasticmodels which characterize the randomness in the arrival anddeparture process-- capacity itself is a random variable whichcan vary from cycle to cycle, and fixed-time equivalent modelswhich treat actuated control as equivalent pretimed modelsoperating at the average cycle and average splits.

Finally, there is a short discussion of concepts related to adaptivesignal control schemes such as the MOVA systems in the UnitedKingdom and OPAC in the U.S. Because these approachesfocus primarily on optimal signal control ratherthan performance modeling, they are somewhat beyond thescope of this document.

There are many areas in traffic signal performance that deserve

overflow queue at a signal is an area that is not well understood

have been documented by Prosser and Dunne (1994) and


9 - 30

References

Akçelik, R. (1980). Time-Dependent Expressions for Delay, Brookes, D. and M. G. Bell (1991). Expected Delays and StopStop Rate and Queue Length at Traffic Signals. Australian Calculation for Discrete Adaptive Traffic Signal Control.Road Research Board, Internal Report, AIR 367-1. Proceedings, First International Symposium on Highway

Akçelik, R. (1988). The Highway Capacity Manual DelayFormula for Signalized Intersections. ITE Journal 58(3),pp. 23-27. Delays. Traffic Engineering and Control, 18(11),

Akçelik, R. and N. Rouphail (1993). Estimation of Delays atTraffic Signals for Variable Demand Conditions.Transportation Research-B, Vol. 27B, No. 2, pp. 109-131.

Akçelik, R. (1994). Estimation of Green Times and Cycle Time for Vehicle-Actuated Signals. Transportation Research

Board 1457, pp.63-72.Akçelik, R. and E. Chung (1994). Traffic Performance

Models for Unsignalized Intersections and Fixed-TimeSignals. Proceedings, Second International Symposium onHighway Capacity, Sydney, Australia, ARRB, Volume I, pp. 21-50.

Akçelik, R. and N. Rouphail (1994). Overflow Queues andDelays with Random and Platoon Arrivals at SignalizedIntersections, Journal of Advanced Transportation, Volume28(3), pp. 227-251.

Akçelik, R. (1995a). Extension of the Highway CapacityManual Progression Factor Method for Platooned Arrivals,Research Report ARR No. 276, ARRB Transport ResearchLtd., Vermont South, Australia.

Akçelik, R. (1995b). Signal Timing Analysis for Vehicle Actuated Control. Working Paper WD TE95/007, ARRBTransport Research Ltd., Vermont South, Australia.

Akçelik, R. and E. Chung (1995). Calibration of PerformanceModels for Traditional Vehicle-Actuated and Fixed-TimeSignals. Working Paper WD TO 95/103, ARRB TransportResearch Ltd., Vermont South, Australia.

Allsop, R. (1972). Delay at a Fixed Time Traffic Signal-I: Theoretical Analysis. Transportation Science, 6,

pp. 260-285.Beckmann, M. J., C. B. McGuire, and C. B. Winsten (1956).

Studies in the Economics in Transportation. New Haven,Yale University Press.

Bell, M. G. H. and D. Brookes (1993). Discrete Time-Adaptive Traffic Signal Control. Transportation Research,Vol. 1C, No. 1, pp. 43-55.

Brilon, W. and N. Wu (1990). Delays At Fixed-time TrafficSignals Under Time Dependent Traffic Conditions. TrafficEngineering and Control, 31(12), pp. 623-63.

Capacity, Karlsruhe, Germany, U. Brannolte, Ed.Catling, I. (1977). A Time-Dependent Approach To Junction

pp. 520-523,526.Clayton, A. (1941). Road Traffic Calculations. J. Inst. Civil.

Engrs, 16, No.7, pp.247-284, No. 8, pp. 588-594.Courage, K. G. and P. P. Papapanou (1977). Estimation of

Delay at Traffic-Actuated Signals. Transportation ResearchRecord, 630, pp. 17-21.

Courage, K. G., C. E. Wallace, and R. Alqasem (1988).Modeling the Effect of Traffic Signal Progression on Delay.Transportation Research Record, 1194, TRB, NationalResearch Council, Washington, DC,pp. 139-146.

Cowan, R. (1978). An Improved Model for Signalized Intersections with Vehicle-Actuated Control. J. Appl. Prob.

15, pp. 384-396.Cronje, W. B. (1983a). Derivation of Equations for Queue

Length, Stops, and Delays for Fixed Time Traffic Signals.Transportation Research Record, 905, pp. 93-95.

Cronje, W. B. (1983b). Analysis of Existing Formulas for Delay, Overflow, and Stops. Transportation Research

Record, 905, pp. 89-93.Darroch, J. N. (1964a). On the Traffic-Light Queue. Ann.

Math. Statist., 35, pp. 380-388.Darroch, J. N., G. F. Newell, and R. W. J. Morris (1964b).

Queues for a Vehicle-Actuated Traffic Light. OperationalResearch, 12, pp. 882-895.

Dunne, M. C. (1967). Traffic Delay at a SignalizedIntersection with Binomial Arrivals. TransportationScience, 1, pp. 24-31.

Fambro, D. B., E. C. P. Chang, and C. J. Messer (1991).Effects of the Quality of Traffic Signal Progression onDelay. National Cooperative Highway Research ProgramReport 339, Transportation Research Board, NationalResearch Council, Washington, DC.

Gartner, N. (1982-1983). Demand Responsive DecentralizedUrban Traffic Control. Part I: Single Intersection Policies;Part II: Network Extensions, Office of University Research,U.S.D.O.T.

Gazis, D. C. (1974). Traffic Science. A Wiley-Intersection Publication, pp. 148-151.


9 - 31

Gerlough, D. L. and M. J. Huber (1975). Traffic Flow Theory. Lin, F. (1992). Modeling Average Cycle Lengths and Green Transportation Research Board Special Report, 165,

National Research Council, Washington, DC.Haight, F. A. (1959). Overflow At A Traffic Flow.

Biometrika. Vol. 46, Nos. 3 and 4, pp. 420-424.Haight, F. A. (1963). Mathematical Theories of Traffic Flow.

Academic Press, New York.Henry, J., J. Farges, and J. Tuffal (1983). The PRODYN

Real-Time Traffic Algorithm. 4th IFAC-IFIC--IFORS onControl in Transportation Systems, Baden-Baden.

Hillier, J. A. and R. Rothery (1967). The Synchronization ofTraffic Signals for Minimum Delays. TransportationScience, 1(2), pp. 81-94.

Hutchinson, T. P. (1972). Delay at a Fixed Time Traffic Signal-II. Numerical Comparisons at Some Theoretical

Expressions. Transportation Science, 6(3), pp. 286-305.Kimber, R. M. and E. M. Hollis (1978). Peak Period Traffic

Delay at Road Junctions and Other Bottlenecks. TrafficEngineering and Control, Vol. 19, No. 10, pp. 442-446.

Kimber, R. and E. Hollis (1979). Traffic Queues and Delaysat Road Junctions. TRRL Laboratory Report, 909, U.K.

Li, J., N. Rouphail, and R. Akçelik (1994). Overflow DelayEstimation for Intersections with Fully-Actuated Signal Australian Road Research Board Bulletin No.4, (SupersededControl. Presented at the 73rd Annual Meeting of TRB,Washington, DC. Miller, A. J. (1968b). The Capacity of Signalized

Lighthill, M. H. and G. B. Whitham (1957). On Kinematic Intersections in Australia. Australian Road Research Board,Waves: II. A Theory of Traffic Flow on Long CrowdedRoads. Proceedings of the Royal Society, London SeriesA229, No. 1178, pp. 317-345. Control by Vehicle Actuated Signals. Traffic Engineering

Little, J. D. C. (1961). Approximate Expected Delays for Several Maneuvers by Driver in a Poisson Traffic.

Operations Research, 9, pp. 39-52.Lin, F. B. (1982a). Estimation of Average Phase Duration of

Full-Actuated Signals. Transportation Research Record,881, pp. 65-72.

Lin, F. B. (1982b). Predictive Models of Traffic-Actuated Cycle Splits. Transportation Research - B, Vol. 16B,

No. 5, pp. 361-372.Lin, F. (1989). Application of 1985 Highway Capacity Manual for Estimating Delays at Signalized Intersection.

Transportation Research Record, 1225, TRB, NationalResearch Council, Washington DC, pp. 18-23.

Lin, F. (1990). Estimating Average Cycle Lengths and GreenIntervals of Semiactuated Signal Operations for Level-of-Service Analysis. Transportation Research Record, 1287,TRB, National Research Council, Washington DC,pp. 119-128.

Intervals of Semi-actuated Signal Operations with ExclusivePedestrian-actuated Phase. Transportation Research, Vol.26B, No. 3, pp. 221-240.

Lin, F. B. and F. Mazdeyasa (1983). Delay Models of TrafficActuated Signal Controls. Transportation Research Record,905, pp. 33-38.

Luh, J. Z. and Chung Yee Lee (1991). Stop Probability andDelay Estimations at Low Volumes for Semi-ActuatedTraffic Signals. Transportation Science, 25(1), pp. 65-82.

May, A. D. and H. M. Keller (1967). A DeterministicQueuing Model. Transportation Research, 1(2),pp. 117-128.

May, A. D. (1990). Traffic Flow Fundamentals. Prentice Hall, Englewood Cliffs, New Jersey, pp. 338-375.McNeil, D. R. (1968). A Solution to the Fixed-Cycle Traffic

Light Problem for Compound Poisson Arrivals. J. Appl.Prob. 5, pp. 624-635.

Miller, A. J. (1963). Settings for Fixed-Cycle Traffic Signals.Operational Research Quarterly, Vol. 14, pp. 373-386.

Miller, A. J. (1968a). Australian Road Capacity Guide - Provisional Introduction and Signalized Intersections.

by ARRB report ARR No. 123, 1981).

ARRB Bulletin No.3.Morris, R. W. T. and P. G. Pak-Boy (1967). Intersection

and Control, No.10, pp. 288-293.Newell, G. F. (1960). Queues for a Fixed-Cycle Traffic Light.

The Annuals of Mathematical Statistics, Vol.31, No.3,pp. 589-597.

Newell, G. F. (1965). Approximation Methods for Queues withApplication to the Fixed-Cycle Traffic Light. SIAMReview, Vol.7.

Newell, G. F. (1969). Properties of Vehicle Actuated Signals:I. One-Way Streets. Transportation Science, 3, pp. 30-52.

Newell, G. F. (1989). Theory of Highway Traffic Signals.UCB-ITS-CN-89-1, Institute of Transportation Studies,University of California.

Newell, G. F. (1990). Stochastic Delays on Signalized Arterial Highways. Transportation and Traffic Theory,

Elsevier Science Publishing Co.,Inc., M. Koshi, Ed.,pp. 589-598.


9 - 32

Newell, G. F. and E. E. Osuna (1969). Properties of Vehicle- Rouphail, N. and R. Akçelik (1992a). Oversaturation DelayActuated Signals: II. Two-Way Streets. Transportation Estimates with Consideration of Peaking. TransportationScience, 3, pp. 99-125. Research Record, 1365, pp. 71-81.

Ohno, K. (1978). Computational Algorithm for a Fixed Cycle Rouphail, N. and R. Akçelik(1992b). A Preliminary Model ofTraffic Signal and New Approximate Expressions for Queue Interaction at Signalized Paired Intersections,Average Delay. Transportation Science, 12(1), pp. 29-47.

Olszewski, P. S. (1988). Efficiency of Arterial Signal Coordination. Proceedings 14th ARRB Conference, 14(2),

pp. 249-257.Olszewski, P. (1990a). Modelling of Queue Probability Distribution at Traffic Signals. Transportation and Traffic

Flow Theory, Elsevier Science Publishing Co., Inc., M.Koshi, Ed., pp. 569-588.

Olszewski, P. (1990b). Traffic Signal Delay Model for Non-Uniform Arrivals. Transportation Research Record, 1287,pp. 42-53.

Pacey, G. M. (1956). The Progress of a Bunch of VehiclesReleased from a Traffic Signal. Research Note No.Rn/2665/GMP. Road Research Laboratory, London(mimeo).

Potts, R. B. (1967). Traffic Delay at a Signalized Intersectionwith Binomial Arrivals. pp. 126-128.

Robertson, D. I. (1969). TRANSYT: A Traffic Network StudyTool. Road Research Laboratory Report LR 253,Crowthorne.

Rorbech, J. (1968). Determining The Length of The ApproachLanes Required at Signal-Controlled Intersections onThrough Highways. Transportation Research Record, 1225,TRB, National Research Council, Washington, DC, pp.18-23.

Rouphail, N. (1989). Progression Adjustment Factors atSignalized Intersections. Transportation Research Record,1225, pp. 8-17.

Proceedings, 16th ARRB Conference 16(5), Perth, Australia,pp. 325-345.

Stephanopoulos, G., G. Michalopoulos, and G. Stephanopoulos (1979). Modelling and Analysis of TrafficQueue Dynamics at Signalized Intersections.Transportation Research, Vol. 13A, pp. 295-307.

Tarko, A., N. Rouphail, and R. Akçelik (1993b). OverflowDelay at a Signalized Intersection Approach Influenced byan Upstream Signal: An Analytical Investigation.Transportation Research Record, No. 1398, pp. 82-89.

Transportation Research Board (1985). Highway CapacityManual, Special Report 209, National Research Council,Washington, DC.

Transportation Research Board (1993). Highway CapacityManual, Chapter 9 (Signalized Intersections), NationalResearch Council, Washington, DC.

Van As, S. C (1991). Overflow Delay in SignalizedNetworks. Transportation Research - A, Vol. 25A,No. 1, pp. 1-7.

Vincent, R. A. and J. R. Pierce (1988). MOVA: Traffic Responsive, Self Optimizing Signal Control for Isolated

Intersections. TRRL Lab Research Report, 170.Webster, F. V. (1958). Traffic Signal Settings. Road Research Laboratory Technical Paper No. 39, HMSO,

London.Wirasinghe, S. C. (1978). Determination of Traffic Delays

from Shock-Wave Analysis. Transportation Research, Vol.12, pp. 343-348.

President, KLD Associates, Inc. 300 Broadway, Huntington Station, NY 1174618

TRAFFIC SIMULATION

BY EDWARD LIEBERMAN18

AJAY K. RATHI

�2

��

xj(m)x


a = acceleration response of follower vehicle to some stimulusfv = instantaneous speed of lead vehiclev = instantaneous speed of follower vehiclefd = projected maximum deceleration rate of lead vehicled = projected maximum deceleration rate of follower vehiclefR = reaction time lag of driver in follower vehiclefI = ith replicate of specified seed, Si 0R = ith random number Ih = headway separating vehicles (sec)H = mean headway (sec)h = minimum headway (sec)minR = random numberX = ith observation (sample) of an MOE Iμ = mean of samplex

= variance= estimate of variance

t , = upper 1-�/2 critical point of the t distribution with n-1 degrees of freedom n-1 1- � /2

= mean of m observation of jth batch= grand sample mean across batches

N = number of replications of the ith strategyivar(X ) = variance of statistic, X

��

10.TRAFFIC SIMULATION

10.1 Introduction

Simulation modeling is an increasingly popular and effective tool for this purpose as an integral element of the ATMS researchfor analyzing a wide variety of dynamical problems which are and development activity.not amenable to study by other means. These problems areusually associated with complex processes which can not readilybe described in analytical terms. Usually, these processes arecharacterized by the interaction of many system components orentities. Often, the behavior of each entity and the interaction of different geometric designs before the commitment ofa limited number of entities, may be well understood and can be resources to construction.reliably represented logically and mathematically withacceptable confidence. However, the complex, simultaneousinteractions of many system components cannot, in general, beadequately described in mathematical or logical forms.

Simulation models are designed to "mimic" the behavior of suchsystems. Properly designed models integrate these separate statistics provided can form the basis for identifying designentity behaviors and interactions to produce a detailed, flaws and limitations. These statistics augmented withquantitative description of system performance. Specifically, animation displays can provide invaluable insights guidingsimulation models are mathematical/logical representations (or the engineer to improve the design and continue the process.abstractions) of real-world systems, which take the form ofsoftware executed on a digital computer in an experimental 4. Embed in other toolsfashion.

The user of traffic simulation software specifies a “scenario”(e.g., highway network configuration, traffic demand) as modelinputs. The simulation model results describe systemoperations in two formats: (1) statistical and (2) graphical. Thenumerical results provide the analyst with detailed quantitativedescriptions of what is likely to happen. The graphical and CORSIM model within the Traffic Research Laboratoryanimated representations of the system functions can provide (TreL) developed for FHWA; and (5) the simulation moduleinsights so that the trained observer can gain an understanding of the EVIPAS actuated signal optimization program.of why the system is behaving this way. However, it is theresponsibility of the analyst to properly interpret the wealth of 5. Training personnelinformation provided by the model to gain an understanding of Simulation can be used in the context of a real-timecause-and-effect relationships. laboratory to train operators of Traffic Management Centers.

Traffic simulation models can satisfy a wide range of time traffic control computer, acts as a surrogate for the real-requirements: world surveillance, communication and traffic environments.

1. Evaluation of alternative treatments 6. Safety AnalysisWith simulation, the engineer can control the experimental Simulation models to “recreate” accident scenarios haveenvironment and the range of conditions to be explored. proven to be indispensable tools in the search to build saferHistorically, traffic simulation models were used initially to vehicles and roadways. An example is the CRASH programevaluate signal control strategies, and are currently applied used extensively by NHTSA.

2. Testing new designsTransportation facilities are costly investments. Simulationcan be applied to quantify traffic performance responding to

3. As an element of the design processThe classical iterative design paradigm of conceptual designfollowed by the recursive process of evaluation and designrefinement, can benefit from the use of simulation. Here, thesimulation model can be used for evaluation; the detailed

In addition to its use as a stand-alone tool, simulation sub-models can be integrated within software tools designed toperform other functions. Examples include: (1) the flowmodel within the TRANSYT-7F signal optimization; (2) theDYNASMART simulation model within a dynamic trafficassignment; (3) the simulation component of theINTEGRATION assignment/control model; (4) the

Here, the simulation model, which is integrated with a real-

��

��

This compilation of applications indicates the variety and scope potential. Unlike the other chapters of this monograph, we willof traffic simulation models and is by no means exhaustive. not focus exclusively on theoretical developments -- althoughSimulation models can also be supportive of analytical modelssuch as PASSER, and of computational procedures such as theHCS. While these and many other computerized tools do notinclude simulation sub-models, users of these tools can enhancetheir value by applying simulation to evaluate their performance.

This chapter is intended for transportation professionals,researchers, students and technical personnel who eithercurrently use simulation models or who wish to explore their

fundamental simulation building blocks will be discussed.Instead, we will describe the properties, types and classes oftraffic simulation models, their strengths and pitfalls, usercaveats, and model-building fundamentals. We will emphasizehow the user can derive the greatest benefits from simulationthrough proper interpretation of the results, with emphasis on theneed to adequately calibrate the model and to apply rigorousstatistical analysis of the results.

10.2 When Should the Use of Simulation Models be Considered?

Since simulation models describe a dynamical process in in order to explain why the resulting statistics werestatistical and pictorial formats., they can be used to analyze a produced.wide range of applications wherever...

� Mathematical treatment of a problem is infeasible orinadequate due to its temporal or spatial scale, and/or It must be emphasized that traffic simulation, by itself, cannot bethe complexity of the traffic flow process.

� The assumptions underlying a mathematicalformulation (e.g., a linear program) or an heuristicprocedure (e.g., those in the Highway CapacityManual) cast some doubt on the accuracy orapplicability of the results.

� The mathematical formulation represents the dynamictraffic/control environment as a simpler quasi steady-state system.

� There is a need to view vehicle animation displays togain an understanding of how the system is behaving

� Congested conditions persist over a significant time.

used in place of optimization models, capacity estimationprocedures, demand modeling activities and design practices.Simulation can be used to support such undertakings, either asembedded submodels or as an auxiliary tool to evaluate andextend the results provided by other procedures. Somerepresentative statistics (called Measures of Effectiveness,MOE) provided by traffic simulation models are listed in Table10.1.

Such statistics can be presented for each specified highwaysection (network link) and for each specified time period, toyield a level of detail that is both spatially and temporallydisaggregated. Aggregations of these data, by subnetwork andnetwork-wide, and over specified time periods, may also beprovided.

10.3 Examples of Traffic Simulation Applications

Given the great diversity of applications that are suitable for theuse of traffic simulation models, the following limited number ofexamples provides only a limited representation of pastexperience.

10.3.1 Evaluation of Signal ControlStrategies

This study (Gartner and Hou, 1992) evaluated and compared theperformance of two arterial traffic control strategies,

��

��

Table 10.1 Simulation Output Statistics: Measures of Effectiveness

Measure for Each Link and for Entire Network

Travel: Vehicle-Miles Bus Travel Time

Travel Time: Vehicle-minutes Bus Moving Time

Moving Time: Vehicle-minutes Bus Delay

Delay Time: Vehicle-minutes Bus Efficiency: Moving Time persons-minutes Total Travel Time

Efficiency: Moving Time Bus Speed Total Travel Time

Mean Travel Time per Vehicle-Mile Bus Stops

Mean Delay per Vehicle-Mile Time bus station capacity exceeded

Mean Travel Time per vehicle Time bus station is empty

Mean Time in Queue Fuel consumed

Mean Stopped Time CO Emissions

Mean Speed HC Emissions

Vehicle Stops NOX Emissions

Link Volumes Occupancy

Mean Link Storage Area Consumed

Number of Signal Phase Failures

Average Queue Length

Maximum Queue Length

Lane Changes

Bus Trips

Bus Person Trips

��

��

MULTIBAND and MAXBAND, employing the TRAF- system optimal (SO) equilibrium calculations for a specifiedNETSIM simulation model. The paper describes the statistical network, over a range of traffic loading conditions fromanalysis procedures, the number of simulation replications unsaturated to oversaturated. This is an example of trafficexecuted and the resulting 95 percent confidence intervals, and simulation used as a component of a larger model to perform athe results of the analysis. complex analysis of an ITS initiative.

Figure 10.1 is taken from this paper and illustrates how Figure 10.2 which is taken from the cited paper illustrates howsimulation can provide objective, accurate data sufficient to simulation can produce internally consistent results for largedistinguish between the performance of alternative analytical scale projects, of sufficient resolution to distinguish between twomodels, within the framework of a controlled experiment. comparable equilibrium assignment approaches.

10.3.2 Analysis of Equilibrium DynamicAssignments (Mahmassani andPeeta, 1993)

This large-scale study used the DYNASMART simulation-assignment model to perform both user equilibrium (UE) and

10.3.3 Analysis of Corridor DesignAlternatives (Korve Engineers1996)

This analysis employed the WATSim simulation model toevaluate alternative scenarios for increasing capacity andimproving traffic flow on a freeway connection, SR242, in

Figure 10.1 Average Delay Comparison, Canal Street,

MULTIBAND & MAXBAND (KLD-242).

��

��

Figure 10.2Comparison of average trip times (minutes) of SO & UE...(KLD-243).

California and ensuring a balanced design relative to freeway could mitigate the extent and duration of congestion within theSR4 on the north and I680 to the south. Design alternatives area, thereby improving performance and productivity. Here, aconsidered for three future periods (years 2000, 2010, 2020) new control concept was tested on a real-world test-bed: aincluded geometric changes, widening, HOV lanes and ramp section of Manhattan. This example illustrates the value inmetering. Given the scale of this 20-mile corridor and the strong testing new “high risk” ideas with simulation without exposinginteractions of projected design changes for the three highways, the public to possible adverse consequences, and prior tothe use of simulation provided a statistical basis for quantifying expending resources to implement these concepts.the operational performance of the corridor sections for eachalternative. These examples certainly do not represent the full range of traffic

This example illustrates the use of simulation as an element of of traffic simulation in the areas of (1) traffic control; (2)the design process with the capability of analyzing candidate transportation planning; (3) design; and (4) research.designs of large-scale highway systems in a manner that liesbeyond the capabilities of a straight-forward HCM analysis.

10.3.4 Testing New Concepts

The TRAF-NETSIM simulation model was used by Rathi andLieberman (1989) to determine whether the application ofmetering control along the periphery of a congested urban area

simulation applications. Yet, they demonstrate the application

10.4 Classification of SimulationModels

Almost all traffic simulation models describe dynamical systems-- time is always the basic independent variable. Continuoussimulation models describe how the elements of a system change

��

��

state continuously over time in response to continuous stimuli.Discrete simulation models represent real-world systems (thatare either continuous or discrete) by asserting that their stateschange abruptly at points in time. There are generally two typesof discrete models:

� Discrete time� Discrete event

The first, segments time into a succession of known timeintervals. Within each such interval, the simulation modelcomputes the activities which change the states of selectedsystem elements. This approach is analogous to representing aninitial-value differential equation in the form of a finite-difference expression with the independent variable, �t.

Some systems are characterized by entities that are "idle" muchof the time. For example, the state of a traffic signal indication(say, green) remains constant for many seconds until its statechanges instantaneously to yellow. This abrupt change in stateis called an event. Since it is possible to accurately describe theoperation of the signal by recording its changes in state as asuccession of [known or computed] timed events, considerablesavings in computer time can be realized by only executing theseevents rather than computing the state of the signal second-by-second. For systems of limited size or those representing entitieswhose states change infrequently, discrete event simulations aremore appropriate than are discrete time simulation models, andare far more economical in execution time. However, forsystems where most entities experience a continuous change instate (e.g., a traffic environment) and where the model objectivesrequire very detailed descriptions, the discrete time model islikely to be the better choice.

Simulation models may also be classified according to the levelof detail with which they represent the system to be studied:

� Microscopic (high fidelity)� Mesoscopic (mixed fidelity)� Macroscopic (low fidelity)

A microscopic model describes both the system entities and theirinteractions at a high level of detail. For example, a lane-changemaneuver at this level could invoke the car-following law for thesubject vehicle with respect to its current leader, then withrespect to its putative leader and its putative follower in thetarget lane, as well as representing other detailed driver decision

processes. The duration of the lane-change maneuver can alsobe calculated.

A mesoscopic model generally represents most entities at a highlevel of detail but describes their activities and interactions at amuch lower level of detail than would a microscopic model. Forexample, the lane-change maneuver could be represented forindividual vehicles as an instantaneous event with the decisionbased, say, on relative lane densities, rather than detailed vehicleinteractions.

A macroscopic model describes entities and their activities andinteractions at a low level of detail. For example, the trafficstream may be represented in some aggregate manner such as astatistical histogram or by scalar values of flow rate, density andspeed. Lane change maneuvers would probably not berepresented at all; the model may assert that the traffic stream isproperly allocated to lanes or employ an approximation to thisend.

High-fidelity microscopic models, and the resulting software, arecostly to develop, execute and to maintain, relative to the lowerfidelity models. While these detailed models possess thepotential to be more accurate than their less detailedcounterparts, this potential may not always be realized due to thecomplexity of their logic and the larger number of parametersthat need to be calibrated.

Lower-fidelity models are easier and less costly to develop,execute and to maintain. They carry a risk that theirrepresentation of the real-world system may be less accurate,less valid or perhaps, inadequate. Use of lower-fidelitysimulations is appropriate if:

� The results are not sensitive to microscopic details.� The scale of the application cannot accommodate the

higher execution time of the microscopic model.� The available model development time and resources

are limited.

Within each level of detail, the developer has wide latitude indesigning the simulation model. The developer must identify thesensitivity of the model's performance to the underlying featuresof the real-world process. For example, if the model is to beused to analyze weaving sections, then a detailed treatment oflane-change interactions would be required, implying the needfor a micro- or mesoscopic model. On the other hand, if the

��

��

model is designed for freeways characterized by limited merging Traffic simulation models have taken many forms depending onand no weaving, describing the lane-change interactions in great their anticipated uses. Table 10.2 lists the TRAF family ofdetail is of lesser importance, and a macroscopic model may be models developed for the Federal Highway Administrationthe suitable choice. (FHWA), along with other prominent models, and indicates their

Another classification addresses the processes represented by the Some traffic simulation models consider a single facilitymodel: (1) Deterministic; and (2) Stochastic. Deterministicmodels have no random variables; all entity interactions aredefined by exact relationships (mathematical, statistical orlogical). Stochastic models have processes which includeprobability functions. For example, a car-following model canbe formulated either as a deterministic or stochastic relationshipby defining the driver's reaction time as a constant value or as arandom variable, respectively.

respective classifications. This listing is necessarily limited.

(NETSIM, NETFLO 1 and 2: surface streets; FRESIM,FREFLO: freeways; ROADSIM: two-lane rural roads;(CORSIM) integrates two other simulation models, FRESIMand NETSIM; INTEGRATION, DYNASMART, TRANSIMSare components of larger systems which include demand modelsand control policies; while CARSIM is a stand-alone simulationof a car-following model. It is seen that traffic simulation modelstake many forms, each of which satisfies a specific area ofapplication.

Table 10.2Representative Traffic Simulation Models

Name Discrete Discrete Micro Mesoscopic Macro Deterministic Stochastic Time Event

NETSIM X X X

NETFLO 1 X X X

NETFLO 2 X X X

FREFLO X X X

ROADSIM X X X

FRESIM X X X

CORSIM X X X

INTEGRATION X X X

DYNASMART X X X

CARSIM X X X

TRANSIMS X X X

��

��

10.5 Building Traffic Simulation Models

The development of a traffic simulation model involves the 4) Calibrate the Modelfollowing activities: - Collect/acquire data to calibrate the model.

1) Define the Problem and the Model Objectives- State the purpose for which the model is being 5) Model Verification

developed. - Establish that the software executes in accord with- Define the information that the model must the design specification.

provide. - Perform verification at the model component level.

2) Define the System to be Studied 6) Model Validation- Disaggregate the system to identify its major - Collect, reduce, organize data for purposes of

components. validation.- Define the major interactions of these - Establish that the model describes the real systemcomponents. at an acceptable level of accuracy over its entire- Identify the information needed as inputs. domain of operation; apply rigorous statistical- Bound the domain of the system to be modeled. testing methods.

3) Develop the Model 7) Documentation- Identify the level of complexity needed to satisfy the - Executive Summary

stated objectives. - Users Manual- Classify the model and define its inputs and - Model documentation: algorithms and software

outputs.- Define the flow of data within the model. The development of a traffic simulation model is not a “single-- Define the functions and processes of the model pass” process. At each step in the above sequence, the analyst

components. must review the activities completed earlier to determine- Determine the calibration requirements and form: whether a revision/extension is required before proceeding

scalars, statistical distributions, further. For example, in step 5 the analyst may verify that theparametric dependencies. software is replicating a model component properly as designed,

- Develop abstractions (i.e., mathematical-logical-statistical algorithms) of eachmajor system component, their activities andinteractions.

- Create a logical structure for integrating thesemodel components to support the flow of dataamong them.

- Select the software development paradigm,programming language(s), user interface, pointless to proceed with validation (step 6) if it is known thatpresentation formats of model results. the verification activity is incomplete.

- Design the software: simulation, structured orobject-oriented programming language; database, Step 3 may be viewed as the most creative activity of therelational/object oriented. development process. The simulation logic must represent all

- Document the logic and all computational relevant interactions by suitably exploring the universe ofprocedures. possibilities and representing the likely outcome. These

- Develop the software code and debug.

- Introduce this data into the model.

but that its performance is at a variance with theoreticalexpectations or with empirical observations. The analyst mustthen determine whether the calibration is adequate and accurate(step 4); whether the model’s logical/mathematical design iscorrect and complete (step 3); whether all interactions with othermodel components are properly accounted for and that thespecified inputs are adequate in number and accuracy (step 2).This continual feedback is essential; clearly, it would be

combinations of interactions are called processes which

Si � (aSi�1�b) mod c.

Ri �Si

c.

h � (H � hmin) [� ln(1�R)]�H�hmin

af � F (vl,vf,s,dl,df,Rf,Pi)

��

��

represent specified functions and utilize component models. Asmall sample of these is presented below.

10.5.1 Car-Following

One fundamental interaction present in all microscopic trafficsimulation models is that between a leader-follower pair ofvehicles traveling in the same lane. This interaction takes theform of a stimulus-response mechanism:

(10.1)

where a , the acceleration (response) of the follower vehicle, isfdependent on a number of (stimulus) factors including:

v , v = Speeds of leader, follower vehicles,l frespectively.

s = Separation distance.d , d = Projected deceleration rates of thel f

leader, follower vehicles, respectively.R = Reaction time of the driver in thef

following vehicle.P = Other parameters specific to the car-i

following model.F(�) = A mathematical and logical formulation

relating the response parameter to thestimulus factors.

This behavioral model can be referenced (i.e., executed) tosupport other behavioral models such as lane-changing,merging, etc.

10.5.2 Random Number Generation

All stochastic models must have the ability to generate randomnumbers. Generation of random numbers has historically beenan area of interest for researchers and practitioners. Beforecomputers were invented, people relied on mechanical devicesand their observations to generate random numbers. Whilenumerous methods in terms of computer programs have beendevised to generate random numbers, these numbers only“appear” to be random. This is the reason why some call thempseudo-random numbers.

The most popular approach for random number generation is the“linear congruential method” which employs a recursiveequation to produce a sequence of random integers S as:

where the integers chosen are defined as,c is the modulus, such that c > 0,a is the multiplier such that 0 < a < c,b is the increment such that 0 < b < m, andS is the starting value or the Seed of the random numbero

generator, such that 0 < S < c.o

The ith random number denoted by R is then generated asi

These random number generations are typically used to generaterandom numbers between 0 and 1. That is, a Uniform (0,1)random number is generated. Random variates are usuallyreferred to as the sample generated from a distribution other thanthe Uniform (0,1). More often than not, these random variatesare generated from the Uniform (0,1) random number. Asimulation usually needs random variates during its execution.Based on the distribution specified, there are various analyticalmethods employed by the simulation models to generate therandom variates. The reader is referred to Law and Kelton(1991) or Roberts (1983) for a detailed treatment on this topic.As an example, random variates in traffic simulation are used togenerate a stream of vehicles.

10.5.3 Vehicle Generation

At the outset of a simulation run, the system is “empty”.Vehicles are generated at origin points, usually at the peripheryof the analysis network, according to some headway distributionbased on specified volumes. For example, the shifted negativeexponential distribution will yield the following expression:

K �

15x60

�N

i�1hi

��

��

where Driver: Aggressiveness; responsiveness to stimuli;h = Headway (sec) separating vehicle emissionsH = Mean headway = 3600/V, where V is the

specified volume, vphh = Specified minimum headwaymin

(e.g., 1.2 sec/veh)R = Random number in the range (0 to 1.0),

obtained from a pseudo-random numbergenerator.

Suppose the specified volume, V (vph), applies for a 15-minuteperiod. If the user elects to guarantee that V is explicitlysatisfied by the simulation model, it is necessary to generate Nvalues of h using the above formula repeatedly, generating a newrandom number each time. Here, N = V/4 is the expectednumber of vehicles to be emitted in 15 minutes. The modelcould then calculate the factor, K:

(10.2)

The model would then multiply each of the N values of h , by K,i

so that the resulting sum of (the revised) h will be exactly 15iminutes, ensuring that the user’s specification of demand volumeare satisfied. However, if K � 1.0, then the resulting distributionof generated vehicles is altered and one element of stochasticity(i.e., the actual number of generated vehicles) is removed. Themodel developer must either include this treatment (i.e.,eq.10.2), exclude it; or offer it as a user option with appropriatedocumentation.

10.5.4 A Representative ModelComponent

Consider two elements of every traffic environment: (1) avehicle and (2) its driver. Each element can be defined in termsof its relevant attributes:

Vehicle: Length; width; acceleration limits; decelerationlimit; maximum speed; type (auto, bus, truck, ...);maximum turn radius, etc.

destination (route); other behavioral and decisionprocesses.

Each attribute must be represented by the analyst, some byscalars (e.g., vehicle length); some by a functional relationship(e.g., maximum vehicle acceleration as a function of its currentspeed); some by a probability distribution (e.g., driver gapacceptance behavior). All must be calibrated.

The driver-vehicle combination forms a model component, orentity. This component is defined in terms of its own elementalattributes and its functionality is defined in terms of theinteractions between these elements. For example, the driver'sdecision to accelerate at a certain rate may be constrained by thevehicle's operational limitations. In addition, this systemcomponent interacts with other model entities representing theenvironment under study, including:

� roadway geometrics� intersection configurations� nearby driver-vehicle entities� control devices� lane channelization� conflicting vehicle movements

As an example, the driver-vehicle entity’s interaction with acontrol device depends on the type and current state of the device(e.g., a signal with a red indication), the vehicle’s speed, itsdistance from stop-bar, the driver’s aggressiveness, etc. It is thedeveloper’s responsibility to design the model components andtheir interactions in a manner that satisfies the model objectivesand is consistent with its fidelity.

10.5.5 Programming Considerations

Programming languages, in the context of this chapter, may beclassified as simulation and general-purpose languages.Simulation languages such as SIMSCRIPT and GPSS/H greatlyease the task of developing simulation software by incorporatingmany features which compile statistics and perform queuing andother functions common to discrete simulation modeling.

��

��

General-purpose languages may be classified as procedural(e.g., FORTRAN, PASCAL, C, BASIC), or object-oriented(e.g., SMALLTALK, C++, JAVA). Object-oriented languagesare gaining prominence since they support the concept ofreusable software defining objects which communicate with oneanother to solve a programming task. Unlike procedurallanguages, where the functions are separated from, and operateupon, the data base, objects encapsulate both data describing itsstate, as well as operations (or “methods”) which can change itsstate and interact with other objects.

While object-oriented languages can produce more reliablesoftware, they require a higher level of programming skill thando procedural languages. The developer should select alanguage which is hardware independent, is supported by themajor operating systems and is expected to have a long life,given the rapid changes in the world of software engineering.Other factors which can influence the language selection processinclude: (1) the expected life of the simulation model; (2) theskills of the user community; (3) available budget (time andresources) to develop and maintain the software; and (4) arealistic assessment of available software development skills.

10.6 An Illustration of Simulation Model Building

Given the confines of this chapter, we will illustrate the modeldevelopment process by presenting the highlights of a sampleproblem, but avoiding exhaustive detail.

1) Define the Problem and Model Objectives - An existing logical design. Note also that the problem is bounded -- nomicroscopic stochastic simulation model of freeway traffic grades or horizontal curves are to be considered at this time.does not consider lane-change operations. It has been See Figure 10.3 for the form of these distributions.determined that this model’s results are unreliable as a result. The purpose of this project is to introduce additional logicinto the model to represent lane-changing operations. Thisaddition should provide improved accuracy in estimatingspeed and delay; in addition it will compute estimates of lanechanges by lane, by vehicle type and by direction (to the leftand to the right).

2) Define the System -

a) A freeway of up to six lanes -- level tangentb) Three vehicle types: passenger car; single-unit truck;

tractor-trailer truckc) Required inputs: traffic volume (varies with time);

distributions of free-speed, of acceptable risk (expressedin terms of deceleration rates if lead vehicle brakes), ofmotivation to change lanes, all disaggregated by vehicletype

d) Drivers are randomly assigned an “aggressiveness index”ranging from 1 (very aggressive) to 10 (very cautious)drawn from a uniform distribution to represent the rangeof human behavior.

It must be emphasized that model development is an iterativeprocess. For example, the need for the indicated inputdistributions may not have been recognized during thisdefinition phase, but may have emerged later during the

3) Develop the Model - Since this lane-change model is to beintroduced into an existing microscopic stochastic model,using a procedural language, it will be designed to utilize theexisting software. The model logic moves each vehicle, eachtime-step, �t, starting with the farthest downstream vehicle,then moving the closest upstream vehicle regardless of laneposition, etc. At time, t , the vehicle states are shown inoFigure 10.4(a).

In developing the model, it is essential to identify theindependent functions that need to be performed and tosegregate each function into a separate software module, orroutine. Figure 10.5 depicts the structure -- not the flow --of the software. This structure shows which routines arelogically connected, with data flowing between them. Someroutines reference others more than once, demonstrating thebenefits of disaggregating the software into functionallyindependent modules.

As indicated in Table 10.3 which presents the algorithm forthe Lane Change Executive Routine in both “StructuredEnglish” or “pseudo-code” and as a flow chart, traffic

��

��

Figure 10.3 Several Statistical Distributions.

��

��

Figure 10.4Vehicle Positions during Lane-Change Maneuver

��

��

Figure 10.5Structure Chart of Simulation Modules

Table 10.3Executive Routine

For each vehicle, I:CALL routine MOTIV to determine whether this driver is “motivated” to change lanes, nowIF so, THEN

CALL routine CANLN to identify which of neighboring lanes (if either) areacceptable as potential target lanes

IF the lane to the right is acceptable, THENCALL routine CHKLC to determine whether a lane-change is feasible, now.

Set flag if so.ENDIFIF the lane to the left is acceptable, THEN

CALL routine CHKLC to determine whether a lane-change is feasible, now.Set flag, if so.

ENDIFIF both lane-change flags are set (lane-change is feasible in either direction), THEN

CALL routine SCORE to determine more favorable target laneELSE IF one lane-change flag is set, THEN

Identify that laneENDIFIF a [favored] target lane exists, THEN

CALL routine LCHNG to execute the lane-changeUpdate lane-change statistics

ELSECALL routine CRFLW to move vehicle within this laneSet vehicle’s process code (to indicate vehicle has been moved this time-step)

ENDIFELSE (no lane-change desired)

CALL routine CRFLW to move vehicle within its current laneSet vehicle’s process code

ENDIFcontinue...

��

��

Figure 10.6Flow Diagram.

��

��

simulation models are primarily logical, rather thancomputational in context. This property reflects the fact thattraffic operations are largely the outcome of driver decisionswhich themselves are [hopefully!] logical in context. As thevehicles are processed by the model logic, they transition fromone state to the next. The reader should reference Table 10.3and Figures 10.3, 10.4, 10.5 and 10.6 to follow the discussiongiven below for each routine. The “subject vehicle” is shown asnumber 19 in Figure 10.4.

Executive: Controls the flow of processing, activating(through CALLs) routines to perform thenecessary functions. Also updates lane-changestatistics.

MOTIV: Determines whether a lane-change is required toposition the subject vehicle for a downstreammaneuver or is desired to improve the vehicle’soperation (increase its speed).

CANLN: Determines whether either or both adjoining lanes (1and 3) are suitable for servicing the subject vehicle.

CHKLC: Identifies vehicles 22 and 4 as the leader andfollower, respectively, in target Lane 1; and vehicles16 and 14, respectively, in target Lane 3. The car-following dynamics between the pairs of vehicles, 19and 22, then 4 and 19 are quantified to assess theprospects for a lane-change to Lane 1. Subsequently,the process is repeated between the pairs of vehicles,19 and 16; then 14 and 19; for a lane-change to Lane3. If the gap is inadequate in Lane 1, causing anexcessive, and possibly impossible deceleration byeither the subject vehicle, 19, or the target follower,4, to avoid a collision, then Lane 3, would beidentified as the only feasible target lane. In any case,CHKLC would identify either Lane 1 or Lane 3, orboth, or neither, as acceptable target lanes at thistime, depending on safety considerations.

SCORE: If both adjoining target lanes are acceptable, then thisheuristic algorithm emulates driver reasoning toselect the preferable target lane. It is reasonable toexpect that the target lane with a higher-speed leaderand fewer vehicles -- especially trucks and buses --would be more attractive. Such reasonablenessalgorithms (which are expressed as “rules” in expert

systems), are also common in models which simulatehuman decisions.

LCHNG: After executing the subject vehicle’s lane-changeactivity, the logic performs some neededbookkeeping:

At time, t + �t, vehicle 19 acts as the leader for bothovehicles 14 and 31, who must “follow” (and areconstrained by) its presence.

At time, t +2�t, the logic asserts that the lane changero(no. 19) has committed to the lane-change and no longerinfluences its former follower, vehicle no. 31. Of course,vehicle 14 now follows the lane-change vehicle, 19.

4) Calibrate the Model - Figures 10.3(b) and (c) aredistributions which represent the outcome of a calibrationactivity. The distribution of free-flow speed is site-specificand can be quantified by direct observation (using pairedloop detectors or radar) when traffic conditions are light --LOS A.

The distribution of acceptable decelerations would be verydifficult to quantify by direct observation -- if not infeasible.Therefore, alternative approaches should be considered. Forexample...

� Gather video data (speeds, distance headway) oflane-change maneuvers. Then apply the car-following model with these data to “back-out” theimplied acceptable decelerations. From a sampleof adequate size, develop the distribution.

� On a more macro level, gather statistics of lane-changes for a section of highway. Execute thesimulation model and adjust this distribution ofacceptable deceleration rates until agreement isattained between the lane-changes executed by themodel, and those observed in the real world. Thisis tenuous since it is confounded by the othermodel features, but may be the best viableapproach.

It is seen that calibration -- the process of quantifyingmodel parameters using real-world data -- is often adifficult and costly undertaking. Nevertheless, it is a

��

��

necessary undertaking that must be pursued withsome creativity and tenacity.

5) Model Verification - Following de-bugging,verification is a structured regimen to provideassurance that the software performs as intended(Note: verification does not address the question, “Arethe model components and their interactionscorrect?”). Since simulation models are primarilylogical constructs, rather than computational ones, theanalyst must perform detailed logical path analyses.

Verification is performed at two levels and generally inthe sequence given below:

� Each software routine (bottom-up testing)� Integration of “trees” (top-down testing)...

When completed, the model developer should beconvinced that the model is performing in accord withexpectations over its entire domain of application.

6) Model Validation - Validation establishes that themodel behavior accurately and reliably represents thereal-world system being simulated, over the range ofconditions anticipated (i.e., the model's "domain").Model validation involves the following activities:

� Acquiring real-world data which, to the extentpossible, extends over the model's domain.

� Reducing and structuring these empirical data so these models. Consequently, it is essential that thethat they are in the same format as the data model be documented for...generated by the model.

� Establishing validation criteria, stating the ease the burden of model application.underlying hypotheses and selecting the statistical � Software maintenance personnel.tests to be applied. � Supervisory personnel who must assess the

� Developing the experimental design of thevalidation study, including a variety of "scenarios"to be examined.

� Performing the validation study:- Executing the model using input data and

calibration data representing the real-worldconditions.

- Performing the hypothesis testing.

� Identifying the causes for any failure to satisfy thevalidation tests and repairing the modelaccordingly.

� Validation should be performed at the componentsystem level as well as for the model as a whole.For example, Figure 10.7 compares the resultsproduced by a car-following model, with field datacollected with aerial photographs. Such facevalidation offers strong assurance that the modelis valid. This validation activity is iterative -- asdifferences between the model results and thereal-world data emerge, the developer must“repair” the model, then revalidate. Considerableskill (and persistence) are needed to successfullyvalidate a traffic simulation model.

7) Documentation - Traffic simulation models, as is thecase for virtually all transportation models, are dataintensive. This implies that users must invest effort indata acquisition and input preparation to make use of

� The end user, to provide a “friendly” interface to

potential benefits of using the model.

10.7 Applying Traffic Simulation Models

Considerable skill and attention to detail must be exercised by simulation-based analysis. The following procedure isthe user in order to derive accurate and reliable results from a recommended:

��

��

Figure 10.7Comparison of Trajectories of Vehicles from Simulation Versus Field Data for Platoon 123.

Identify the Problem Domain

What highway facilities are involved? - Consider expected accuracy and reliability.- Surface streets (grid, arterial, both), freeways, - Consider available budget for data acquisition.

rural roads, toll plaza.What is the traffic environment?

- Autos, trucks, buses, LRT, HOV...- Unsaturated, oversaturated conditions.

What is the control environment?- Signals (fixed time, actuated, computer-

controlled), signs- Route guidance

What is the size of the network and duration of the analysisperiod?

Define the Purpose of the Study

What information is needed from the simulation model?- Statistical: MOE sought, level of detail.

- Pictorial: static graphics, animation.What information is available as input and calibration data?

Are results needed on a relative or absolute basis?

What other functions and tools are involved?- Capacity analysis- Design- Demand modeling- Signal optimization

Is the application real-time or off-line?

Investigate Candidate Traffic Simulation Models

Identify strengths and limitations of each.- Underlying assumptions

��

��

- Computing requirements - Signal timing plans, actuated controller settings.- Availability, clarity, completeness of documentation - Traffic volume and patterns; traffic composition.- Availability, reliability, timeliness of software - Transit schedules

support - Other, as required.

Estimate extent and cost of data collection for calibration and Confirm the accuracy of these data through field observation.input preparation.

Determine whether model features match problem needs. required.

Assess level of skill needed to properly apply model. Identify need for accurate operational traffic data: based on

Determine compatibility with other tools/procedures needed forthe analysis. - Select representative locations to acquire these field

Assess the Need to Use a Traffic Simulation Model

Is traffic simulation necessary to perform the analysis of the flow speeds; acceptable gaps for permitted left-turns,problem? etc.

- Are other tools adequate but less costly? - Accept model default values or other data from the- Are your skills adequate to properly apply literature with great care if data collection is

simulation? infeasible or limited by cost considerations.- Can the data needed by traffic simulation be

acquired?

Is traffic simulation highly beneficial even though not Calibration is the activity of specifying data describing trafficnecessary? operations and other features that are site-specific. These data

- Simulation results can confirm results obtained by may take the form of scalar elemets and of statisticalother tools. distributions that are referenced by the logic of stochastic

- Animation displays needed as a presentation simulation models. While traffic simulation models generallymedium. provide default value which represent average conditions for

If it is determined that traffic simulation is needed/advisable, analyst to quantify these data with field observations to the extentcontinue. practicable rather than to accept these default values.

Select Traffic Simulation Model Model Execution

Relate relevant model attributes to problem needs. The application of a simulation model should be viewed as

Determine which model satisfies problem needs to the greatestextent. Consider technical, and cost, time, available skills andsupport, and risks factors.

Data Acquisition

Obtain reliable records of required information.- Design drawings for geometrics.

Undertake field data collection for input and for calibration, as

model’s sensitivity site-specific features; accuracy requirements.

data.- Collect data using video or other methods as

required: saturation flow rates at intersections; free-

Model Calibration

these calibration data elements, it is the responsibility of the

performing a rigorous statistical experiment. The model mustfirst be executed to initialize its database so that the dataproperly represents the initial state of the traffic environment.This requirement can reliably be realized if the environment isinitially at equilibrium.

Thus, to perform an analysis of congested conditions, the analystshould design the experiment so that the initial state of the trafficenvironment is undersaturated, and then specify the changing

��

��

conditions which, over time, censors the congested state which � Anomolous results (e.g., the creation and growth ofis of interest. Similarly, the final state of the traffic environment queues when conditions are believed to beshould likewise be undersaturated, if feasible. undersaturated) can be examined and traced to valid,

It is also essential that the analyst properly specify the dynamic model deficiencies.(i.e., changing) input conditions which describe the trafficenvironment. For example, if one-hour of traffic is to besimulated, the analyst should always specify the variation indemand volumes -- and in other variables -- over that hour at anappropriate level of detail rather than specifying average,constant values of volume.

Finally, if animation displays are provided by the model, thisoption should always be exercised, as discussed below.

Interpretation of Simulation Results

Quite possibly, this activity may be the most critical. It is theanalyst who must determine whether the model results constitutea reasonable and valid representation of the traffic environmentunder study, and who is responsible for any inferences drawnfrom these results. Given the complex processes taking place inthe real-world traffic environment, the analyst must be alert tothe possibility that (1) the model’s features may be deficient inadequately representing some important process; (2) the inputdata and/or calibration specified is inaccurate or inadequate; (3)the results provided are of insufficient detail to meet the projectobjectives; (4) the statistical analysis of the results are flawed (asdiscussed in the following section); or (5) the model has “bugs”or some of its algorithms are incorrect. Animation displays of extracting the sought information and insights from the mass ofthe traffic environment (if available) are a most powerful tool foranalyzing simulation results. A careful and thorough review ofthis animation can be crucial to the analyst in identifying: Proper output analysis is one of the most important aspects of

� Cause-and-effect relationships. Specifically the originsof congested conditions in the form of growing queuescan be observed and related to the factors that caused it.

incongested behavior; to errant input specifications; or to

If the selected traffic simulation lacks an animation feature or ifquestions remain after viewing the animation, then the followingprocedures may be applied:

� Execute the model to replicate existing real-worldconditions and compare its results with observedbehavior. This “face validation”, which is recommendedregardless of the model selected, can identify model orimplementation deficiencies.

� Perform “sensitivity” tests on the study network byvarying key variables and observing model responses ina carefully designed succession of model executions.

� Plot these results. A review will probably uncover theperceived anomalies.

Table 10.1 lists representative data elements provided by trafficsimulation models. Figure 10.8 shows typical graph displayswhile Figure 10.9 displays a "snapshot" of an animation screen.

Note that all the graphical displays can be accessed interactivelyby the user, thus affording the user an efficient means for

data compiled by the simulation model.

any simulation study. A variety of techniques are used,particularly for stochastic model output, to arrive at inferencesthat are supportable by the output. A brief exposition to outputanalysis of simulation data is paresented next.

10.8 Statistical Analysis of Simulation Data

In most efforts on simulation studies, more often than not a large a heuristic model, which is then programmed on the computer,amount of time and money is spent on model development andcoding, but little on analyzing the output data. Simulationpractitioners therefore have more confidence in their results thanis justified. Unfortunately, many simulation studies begin with

and conclude with a single run of the program to yield ananswer. This is a result of overlooking the fact that a simulationis a sampling experiment implemented on a computer andtherefore needs appropriate statistical techniques for its design,

��

��

Figure 10.8Graphical Displays.

��

��

Figure 10.9Animation Snapshot.

analysis, and implementation. Also, more often than not, output Typical goals of analyzing output data from simulationdata from simulation experiments are auto correlated and experiments are to present point estimates of the measures ofnonstationary. This precludes its analysis using classical effectiveness (MOE) and form confidence intervals around thesestatistical techniques which are based on independent and estimates for one particular system design, or to establish whichidentically distributed (IID) observations. simulated system is the best among different alternate

��

��

configurations with respect to some specified MOE. Point system using independent random number streams acrossestimates and confidence intervals for the MOEs can be obtained replications.either from one simulation run or a set of replications of the

10.9 Looking to the FutureWith the traffic simulation models now mounted on high- - Combining simulation with Artificial Intelligenceperformance PCS, and with new graphical user interfaces (GUI) (AI) software. The simulation can provide thebecoming available to ease the burden of input preparation, it is knowledge base in real-time or generate it inreasonable to expect that usage of these models will continue to advance as an off-line activity.increase significantly over the coming years. - Integrating traffic simulation models with other tools

In addition, technology-driven advances in computers, combined optimization models, GIS, office suites, etc.with the expanding needs of the Intelligent Transportation - Providing Internet access.Systems (ITS) program, suggest that the new applications oftraffic simulation can contribute importantly to this program. � Real-time simulators which replicate the performance ofSpecifically: TMC operations. These simulators must rely upon

� Simulation support systems of Advanced Traffic stimuli to ITS real-time software being tested. SuchManagement Systems (ATMS) in the form of: simulators are invaluable for:

- Off-line emulation to test, refine, evaluate, new real- - Testing new ATMS concepts prior to deployment.time control policies. - Testing interfaces among neighboring TMCs.

- On-line support to evaluate candidate ad-hoc - Training TMC operating personnel.responses to unscheduled events and to advise the - Evaluating different ATMS architectures.operators at the Traffic Management Center (TMC) - Educating practitioners and student.as to the “best” response. -Real- - Demonstrating the benefits of ITS programs to statetime component of an advanced control/guidance and municipal officials and to the public throughstrategy. That is, the simulation model would be a animated graphical displays and virtual reality.component of the on-line strategy software.

such as: transportation demand models, signal

simulation models as “drivers” to provide the real-world

References

Gartner, N.H. and D. L. Hou (1992). Comparative Evaluation Korve Engineers (1996). State Route 242 Widening Project -of Alternative Traffic Control Strategies,” Transportation Operations Analysis Report to Contra Costa TransportationResearch Record 1360, Transportation Research Board. Authority.

Mahmassani, H.S. and S. Peeta (1993). Network Performance Rathi, A.K. and E.B. Lieberman (1989). EffectivenessUnder System Optimal and User Equilibrium Dynamic of Traffic Restraint for a Congested Urban Network: AAssignments: Implications for Advanced Traveler Simulation Study. Transportation Research Record 1232.Information Systems Transportation Research Record 1408,Transportation Research Board.

KINETIC THEORIES

BY PAUL NELSON*

* Professor, Department of Computer Science, Texas A&M University, College Station, TX 77843-3112 �

KINETIC THEORIES

11-1

11 Kinetic Theories

Criticisms and accomplishments of the Prigogine-Herman

kinetic theory are reviewed. Two of the latter are identified as

possible benchmarks, against which to measure proposed

novel kinetic theories of vehicular traffic. Various kinetic

theories that have been proposed in order to eliminate defi-

ciencies of the Prigogine-Herman theory are assessed in this

light. None are found to have yet been shown to meet both of

these benchmarks.

Possible objectives and applications for kinetic theo-

ries of vehicular traffic are considered. One of these is the

traditional application to the development of continuum mod-

els, with the resulting microscopically based coefficients.

However, modern computing power makes it possible to con-

sider computational solution of kinetic equations per se, and

therefore direct applications of the kinetic theory (e.g., the

kinetic distribution function). It is concluded that the primary

applications are likely to be found among situations in which

variability between instances is an important consideration

(e.g., travel times, or driving cycles).

11.1 IntroductionOn page 20 of their well-known monograph on the kinetic theory of vehicular traffic, Prigogine and Herman (1971) summarize possible alternate forms of the relaxation term in their kinetic equation of vehicular traffic. They conclude this discussion by issuing the invitation that “the reader may, if he is so inclined, work out the theory using other forms of the relaxation law.” This invitation to explore alternate kinetic models of vehicular traffic has subsequently been accepted by a number of workers, most notably by Paveri-Fontana (1975) and by Phillips (1979, 1977), and more recently by Nelson (1995a), and by Klar and Wegener (1999). The existence of this variety of kinetic models of vehicular traffic raises the issue of how one chooses between them in any particular ap-plication; more generally there arises the issue of the types of applications for which any kinetic model has a role. In these lights, the primary objective of this chapter is to address ques-tions related to what might reasonably be expected from a good kinetic theory of vehicular traffic.

The approach presented here is substantially influ-enced by the work of Nagel (1996), who gave an excellent review of a variety of types of models of vehicular traffic, including continuum (“hydrodynamic”), car-following and particle hopping (cellular automata) models. In particular, he has emphasized that: i) any model necessarily represents some compromise in terms of its fidelity in describing the reality it is intended to represent; ii) different types of models represent engineering judgements as to the relative importance of reso-lution, fidelity and scale for the particular application at hand. To some extent, this chapter is intended to address similar

issues for kinetic models of vehicular traffic. The status of various kinetic models will also be reviewed, in terms of achieving two objectives that seem appropriate to designate as benchmarks, primarily on the basis that the seminal kinetic model of Prigogine and Herman (1971) has been shown to meet those objectives.

It seems appropriate to view kinetic models as occu-pying a point on the model spectrum that is intermediate be-tween continuum (e.g., hydrodynamic) models and micro-scopic (e.g., car-following or cellular automata) models.1 One of the primary applications of kinetic models is to obtain con-tinuum models in a consistent manner from an underlying microscopic model of driver behavior. (See Nelson (1995b) for further thoughts on the role of kinetic models of vehicular traffic as a bridge from microscopic models to macroscopic models.) However, computing power now has advanced to the point that it is practical to consider computational solution of kinetic equations per se. This opens the door to the realistic possibility of applying kinetic models directly to the simula-tion of traffic flow. This is a qualitatively different situation from that prevailing in the 1960’s, when kinetic models of vehicular traffic were initially proposed by Prigogine, Herman and co-workers. (See Prigogine and Herman, 1971, and works cited therein.)

The specific further contents of this chapter are as follows. In Section 2 below, the status of the Prigogine-

1 The word mesoscopic has come into recent vogue to describe mod-els that are, in some sense, intermediate between macroscopic and microscopic models.

KINETIC THEORIES

11-2

Herman (1971) kinetic model of vehicular traffic is reviewed. The intent of this review is to provide an evenhanded discus-sion of both the deficiencies and signal accomplishments of this seminal kinetic theory of vehicular traffic. Two of these accomplishments are suggested as benchmarks that should minimally be met by any proposed novel kinetic model of vehicular traffic. In Section 3 alternate kinetic models that have been proposed in the literature are assessed against these benchmarks, and none are found that yet have been shown to meet both of them. Both of these benchmarks relate to the equilibrium solutions of the Prigogine-Herman kinetic equa-tion, and one of them relates to the recent result of Nelson and Sopasakis (1998) to the effect that under certain circumstances – particularly for sufficiently congested traffic – the Prigog-ine-Herman model admits a two-parameter family of equilib-rium solutions, as opposed to the one-parameter (density) fam-ily that would be expected classically.

In Section 4, the role of kinetic equations as a bridge from microscopic to continuum models is considered. Section 5 is devoted to consideration of the potential applications of the solution of kinetic equations per se.

11.2 Status of the Prigogine-Herman Ki-netic Model

The kinetic model of Prigogine and Herman (1971) is summa-rized in Subsection 2.1. A number of published criticisms of this model, along with alternative models that have been sug-gested to overcome some of these criticisms, are reviewed in Subsection 2.2. In Subsection 2.3 two significant accom-plishments of the Prigogine-Herman theory are described, and suggested as benchmarks against which novel kinetic theories of vehicular traffic should be measured.

11.2.1 The Prigogine-Herman Model The kinetic equation of Prigogine and Herman is

.)1)((0 fPvvcT

ffxfv

tf

�

��

��

(11. 1)

(1)

Here the various symbols have the following meanings:

a) the zero order moment of f(x,v,t),

is vehicular density. b) the ratio of the first and zero order moments,

is mean vehicular speed; c) P is passing probability; d) T is the relaxation time; e) f0 is the corresponding density function for the desired

speed of vehicles; f) f is the density function for the distribution of vehicles in

phase space, so that

is the expected number of vehicles at time t that have po-sition between 1x and 2x and speed between 1v and

2v ( 21 xx � and 21 vv � ).The second term on the left-hand side of Eq. (1), the

streaming term, represents the rate of change of the density function due to motion of the traffic stream, absent any changes of velocity by vehicles. The first term on the right-hand side, which is often called the relaxation term, is the contribution to this rate of change that stems from changes of vehicular speed associated with passing or other causes of acceleration. The second term on the right-hand side, the slowing-down term, stems from deceleration of vehicles that overtake slower-moving vehicles. The relaxation term is phe-nomenological in nature, in that it is based on the underlying assumption that increases in vehicular speed cause the actual density to “relax” toward the desired density with some char-acteristic time T. By contrast, the slowing-down term can be obtained from basic physical arguments, albeit with idealized assumptions such as instantaneous deceleration, treatment of vehicles as point particles (i.e., neglect of the positive length of vehicles), and the validity of what Paveri-Fontana (1975) terms vehicular chaos. The validity of both of these particular forms of the rates of change due to changes of speed has been

,),,(),(0 �

� dvtvxftxc

2

1

2

1

),,(v

v

x

x

dxdvtvxf

),(),,(),(0

txcdvtvxvftxv �

�

KINETIC THEORIES

11-3

questioned, as will be briefly discussed in the following sub-section.

A kinetic equation generally is an equation that in principle, subject to appropriate initial and boundary condi-tions, can be solved for the density function f, as defined above. Some kinetic equations that are alternatives to that of Prigogine and Herman are discussed in Section 3 following

11.2.2 Criticisms of the Prigogine-Herman Model.

The first published serious critique of the Prigogine-Herman kinetic equation seems to be the work of Munjal and Pahl (1969). These workers raise a number of questions,2 but the most fundamental of these fall into one of the following two categories:1. The validity of the slowing-down term (denoted the “in-

teraction term” by these authors) is doubtful in the pres-ence of “queues” (or “platoons”) of vehicles. This stems from the fact that the correlation inherent in platoons in-validates the assumption of vehicular chaos (Paveri-Fontana, 1975), which assumption underpins the particu-lar form of the slowing-down term in the Prigogine-Herman kinetic equation.

2. The absence of a derivation of the relaxation term from first principles raises general questions regarding its va-lidity. The validity of the specific expression (in terms of c) used by Prigogine et al. for the relaxation time T hastherefore “not been proven.” Further, it is therefore also difficult to “conceive the meaning of the relaxation time” and therefore “define a method for its experimental de-termination.”

In addition to noting the first of these concerns, Paveri-Fontana (1975) argues forcefully that it is fundamentally in-correct to treat the desired speed as a parameter, as in done in the Prigogine-Herman kinetic equation. Rather, he suggests the desired speed must be taken as an additional independent

2 Other concerns relate to: i) The necessity to include time depend-ence in the desired speed distribution, owing to the normalization

�max

00 ),(),,(

v

txcdvtvxf ; and ii) the interpretation and func-

tional dependence of the passing probability, P.

variable, on the same footing as the actual speed, and he pro-vides a modification of the Prigogine-Herman equation that accomplishes precisely that.

Prigogine and Herman (1971, Section 3.4, and 1970) dispute the claim of Munjal and Pahl (1969) to the effect that “the validity of the interaction term (i.e., the Prigogine-Herman slowing-down term) is limited to traffic situations where no vehicles are queuing” (parenthetical clarification added). Current opinion seems inclined to agree with Paveri-Fontana (1975) that on balance Munjal and Pahl have the bet-ter of this particular discussion. However, traffic on arterial roads, for which signalized intersections necessarily enforce the formation of platoons, is the only situation that seems thus to be definitely excluded from the domain of the Prigogine-Herman kinetic equation. In particular, it is not a priori clearthat the same objection is valid for the stop-and-go traffic that seems to characterize congested traffic on freeways. Nelson (1995a) has noted that a correlation model is generally needed to obtain a kinetic equation, and vehicular chaos is simply one instance of a correlation model. Other correlation models, which would lead to a kinetic equation other than that of Prigogine and Herman, conceivably could better treat pla-toons, at least under restricted circumstances. Approaches (e.g., Prigogine and Andrews, 1960; Beylich, 1979), in which multiple-vehicle density functions appear as the unknowns to be determined, also offer the potential ability to treat queues within the spirit of the kinetic theory.

Nelson (1995a) introduced the concepts of a mechani-cal model and a correlation model as the fundamental ingredi-ents of any kinetic equation. This work was motivated pre-cisely by the desire to obtain forms of the speeding-up term that are based upon at least the same level of first principles as the classical derivations of the Prigogine-Herman slowing-down term. Klar and Wegener (1999) used this approach to obtain a kinetic equation for traffic flow that accounts for the spatial extent of vehicles. The treatment of vehicles as “points” of zero length is an idealization underlying the Prigogine-Herman kinetic equation that seems not to have been extensively discussed in the earlier literature on traffic flow. Klar and Wegener (1999) show that including the length of vehicles has a significant quantitative effect upon the value of some coefficients in associated continuum models. The observational measurements of the relaxation time by Edie, Herman and Lam (1980) also bear mentioning.

The arguments of Paveri-Fontana (1975) that the de-sired speed must appear as an independent variable in any kinetic equation, so that the density function depends upon the

KINETIC THEORIES

11-4

desired speed, as well as position, actual speed and time, seem to be quite convincing. In order to avoid this complexity, some workers (e.g., Nelson, 1995a) choose to focus upon models in which all drivers have the same desired speed. Paveri-Fontana (1975) represents his modification of the Prigogine-Herman equation, to include desired speed as an independent variable, as valid only for dilute traffic. How-ever, as suggested above, it is not completely clear that this restriction is required, unless the dense traffic also includes a significant fraction of the vehicles in platoons.

11.2.3 Accomplishments of the Prigogine-Herman Model

In view of the deficiencies chronicled in the preceding subsec-tion, why would anyone deem the Prigogine-Herman kinetic equation to be of any interest? That question is answered in this subsection, by describing two significant results that stem from the Prigogine-Herman model.

First, Prigogine and Herman (1971, Chap. 4) demon-strated, under the somewhat restrictive assumption that there exist drivers desiring arbitrarily small speeds, that one can obtain traffic stream models (fundamental diagrams, speed/density relations), say q=Q(c), from the equilibrium solutions (i.e., the solutions that are independent of space and time) of their kinetic equation. (Here q is vehicular flow, and c is, as above, vehicular density.) The procedure is precisely analogous to that giving rise to the Maxwellian distribution and the ideal gas law, when applied to the Boltzmann equation of the kinetic theory of gases. Further, at high concentrations the equilibrium solution is bimodal; that is, it displays two (local) maxima in speed, in qualitative agreement with the observations of Phillips (1977, 1979). (See the following sec-tion for more details of these works.) One of these modes corresponds to a modification of the distribution of desired speeds, and the other (under the assumptions of Prigogine and Herman) to platoon flow in the rather extreme case of stopped traffic (i.e., zero speed). This “multiphase” aspect of con-gested traffic flow has subsequently been rediscovered by a number of workers. Note that this approach gives rise to a traffic stream model from an underlying microscopic model, via the equilibrium solution of a corresponding kinetic equa-tion. Such a theoretical development contrasts with statements sometimes encountered to the effect that traffic stream models must be based upon observational data.

More recently, Nelson and Sopasakis (1998) showed that if one relaxes the assumption of Prigogine and Herman that there exist drivers having arbitrarily small desired speeds, then at sufficiently high densities the equilibrium solution is a two-parameter family. This contrasts with the one-parameter (typically taken as density) family that occurs at low densities, even at all densities under the restrictive assumption of Prigogine and Herman (1971).3

The consequence of the equilibrium solutions of Nel-son and Sopasakis (1998) for the attendant traffic stream model will now be briefly described. Let

where w- and w+ are respectively the lower and upper bound on the desired speeds. Then there exists a positive critical density, denoted ccrit, and defined as the unique root (in c) of the equation F(0;c) = cT(1-P), such that the dependence of mean speed upon density is as follows. Let !*=!*(c) be the unique root (in ! ) of F(!;c) = cT(1-P). If 0� c�ccrit , then !*�0, and the mean speed is given by

However, if c>ccrit , then !*>0, and the mean speed is given by

where now ! can take on any value such that 0 � ! �min{!*,w-}. The parameter ! is the speed of the embedded collective flow, and the preceding equation for the mean speed shows that the overall mean speed increases with increasing speed of the embedded collective flow. Figure 11.1 shows a three-dimensional graphical representation of the resulting “traffic stream model,” for a particular hypothetical desired

3 In some of their work, Prigogine and Herman (1971, Section 4.4, esp. Fig. 4.8 and the related discussion) did permit posi-tive lower bounds for the set of desired speeds, but for reasons that seem unclear at this point their attendant analysis did not identify the full two-parameter range of equilibrium solutions at higher densities.

�

�w

w

dvvc

vfcF ,

)()(

:);( 0

!!

.)1(

1)( *!�

��PcT

cvv

,)1(

1),( !! �

��PcT

cvv

KINETIC THEORIES

11-5

Figure 11-1 Dependence of the mean speed upon density normalized to jam density, "=c/cP, for jam density cP = 200 vpm, P=1-", T=�"/(1-"), with �=0.003 hours, and a uniform desired speed distribution from 40 to 80 mph.

speed distribution. See Nelson and Sopasakis (1998) for more details.

The significance of this three-dimensional presenta-tion of a traffic stream model lies in the fact that it is consis-tent with the well-known tendency (e.g., Drake, Schofer and May, 1967) for traffic flow data to be widely scattered at high densities. The effort to explain this tendency has spawned a number of theories (e.g., Ceder, 1976; Hall, 1987; Disbro and Frame, 1989). The explanation in terms of an embedded col-lective flow seems possibly preferable to these, in that it de-rives from the kinetic theory, which is a well-known branch of traffic flow theory, as opposed to requiring some novel ad hoc theory.

Thus, the Prigogine-Herman kinetic equation has equilibrium solutions that both reproduce the observed bi-modal distribution of speeds at high densities, and provide traffic stream models that reproduce qualitatively the well-known result that at sufficiently high densities mean speeds and flows do not depend exclusively upon vehicular density.

One certainly can envision more ambitious objectives for a kinetic theory of vehicular traffic than these two. Some possi-ble such objectives are discussed further in Sections 4 and 5 below. However, given that the seminal Prigogine-Herman kinetic equation of vehicular traffic does accomplish at least these objectives, it seems appropriate to suggest them as minimal benchmarks that should be met by any alternative kinetic equations that might be proposed. In the following section some of the alternative kinetic equations that have been proposed, as described in the preceding subsection, are assessed against these benchmarks.

11.3 Other Kinetic Models Both benchmarks suggested in the preceding subsection have to do with the equilibrium solutions of the kinetic equation of interest. The equilibrium solutions of the Paveri-Fontana (1975) generalization of the Prigogine-Herman kinetic equa-tion, as described in Subsection 2.2, do not seem to have been definitively ascertained. Indeed, Helbing (1996), who has

−80−60−40−2002040

00.5

1

0

10

20

30

40

50

60

speed of collective flow (mph)

normalized concentration

Mea

n sp

eed

(mph

)

KINETIC THEORIES

11-6

extensively applied the Paveri-Fontana kinetic model in his recent works on the kinetic theory of vehicular traffic, states, in regard to these equilibrium solutions, that “unfortunately it seems impossible to find an analytical expression ….” He then indicates that “empirical data and microsimulations” sug-gest these equilibrium solutions are “approximately a Gaus-sian.” Note that Gaussians are not bimodal. Thus, the Paveri-Fontana model does not seem to have been shown to satisfy either of the benchmarks suggested in the preceding subsec-tion.

Phillips (1977, 1979) develops yet another kinetic equation that is an alternative to the original Prigogine-Herman kinetic model. However, this development seems predicated on a form of the corresponding equilibrium solu-tion that ignores the considerations that led Prigogine and Herman to the “lower mode” of their bimodal equilibrium solution; cf. Eq. (4) of Phillips, 1979. Phillips compared (sketchily in Phillips, 1979, but exhaustively in Phillips, 1977) the equilibrium solution of his kinetic model against measured speed distributions. With one possibly important exception, the agreement seems reasonable. One therefore expects good agreement between the traffic stream model obtained theoreti-cally from the equilibrium solution and that obtained observa-tionally, although Phillips does not explicitly effect such com-parisons. The exception is that a large amount of the data in-dicates a bimodal equilibrium solution; cf. Figs. 3 and 4 of Phillips, 1979, and numerous figures in Phillips, 1977. Thus, although the bimodal nature of an equilibrium solution is missed by the theoretical analysis, it is supported by the asso-ciated observations. In summary, it seems likely that the ki-netic equation of Phillips (1979, 1977) meets the first bench-mark suggested in the preceding section, and possible that a mathematical reassessment of its equilibrium solutions will reveal that it meets the second of these benchmarks. How-ever, neither of these conclusions has yet been conclusively established.

Nelson (1995a) obtained a specific kinetic equation for purposes of providing a concrete illustration of his pro-posed general methodology for obtaining speeding-up (and slowing-down) terms based on first principles (i.e., appropri-ate mechanical and correlation models). In subsequent work (Nelson, Bui and Sopasakis, 1997) it was shown that this ki-netic equation provides a theoretical traffic stream model that agrees well with classical traffic stream models, except near jam density. It has further been shown (Bui, Nelson and So-pasakis, 1996) that a simple modification of the underlying correlation model removes the incorrect behavior near jam density. Thus, this kinetic equation has been shown to meet

the first of the benchmarks suggested in the preceding section. However, the equilibrium solutions of the kinetic equation of Nelson (1995a) are such that it clearly does not meet the sec-ond benchmark (i.e., does not predict scattered flow data un-der congestion). It is possible that the underlying mechanical model could be modified to attain this objective, but that has not been demonstrated.

Klar and Wegener (1999) use numerical techniques to obtain equilibrium solutions of their kinetic equations. They do not explicitly present corresponding traffic stream models. Their numerical equilibrium solutions do not display two modes. It might be difficult to obtain the lower mode, which typically appears as a delta function, by a strictly nu-merical approach.

Table I summarizes the status of the various kinetic models mentioned here, as regards their ability to meet the two benchmarks delineated in Subsection 2.3.

Table 11-I Status of various kinetic models with respect to the benchmarks of Subsection 11.2.3

Benchmark Kinetic Model

Bimodal equilib-rium solutions?

Equilibrium with scattered flows at high densities?

Prigogine-Herman (1971) yes yesPaveri-Fontana (1975) ? ? Phillips (1977, 1979) no no Nelson (1995a) yes noKlar-Wegener (1999) no? no

11.4 Continuum Models from Kinetic Equa-tions

Continuum models historically have played an important role in traffic flow theory. They have been obtained either by sim-ply writing them as analogs of some corresponding fluid dy-namical system (e.g., Kerner and Konhäuser, 1993), or by rational developments from some presumably more basic mi-croscopic (e.g., car-following) model of traffic flow. In the latter case the continuum equations can be developed either directly from the underlying microscopic model that serves as the starting point, or a kinetic model can play an intermediary role between the microscopic and continuum models. For early examples of the former approach, through the steady-state solutions of car-following models, see numerous refer-ences cited in Nelson, 1995b. Nagel (1998) presents a more modern approach, through appropriate formal (“fluid-dynamical”) limits of particle-hopping models.

KINETIC THEORIES

11-7

# $ ,)()( ��

��

��

��

��

��

xccD

xcQ

xtc

Here the primary interest is, of course, in approaches to continuum models that use a kinetic intermediary to the underlying microscopic model. Such approaches often (e.g., Helbing, 1995) follow the route of first taking the first few (one or two) low-order polynomial moments of the kinetic equation, then achieving closure via ad hoc approximations. An alternate approach, via certain formal asymptotic expan-sions (e.g., Hilbert or Chapman-Enskog expansions) is often used in the kinetic theory of gases (e.g., Grad, 1958). In this approach, the number of polynomial moments of the kinetic equation that are taken tend to be determined by the number of invariants that are defined by the dynamics of the microscopic model of the interaction between the constituent “particles” (vehicles, for traffic flow) of the system. This approach leads to a hierarchy of continuum models (e.g., the Euler/Navier-Stokes/Burnett/super-Burnett equations of fluid dynamics), as opposed to the single continuum equation that tends to result from formal limits of microscopic models. At all levels of this hierarchy the parameters of the resulting continuum model are expressed in terms of those of the underlying microscopic model.

Nelson and Sopasakis (1999) applied the Chapman-Enskog expansion to the Prigogine-Herman (1971) kinetic equation. In the region below the critical density described in Subsection 11.2 the lowest (zero) order expansion was found to be a Lighthill-Whitham (1955) continuum model, with as-sociated traffic stream model corresponding to the one-parameter family of equilibrium solutions. The next highest (first-order) solution was found to be a diffusively corrected Lighthill-Whitham model,

(11.2)

where now both the flow function Q(c) and the diffusion coef-ficient D(c) are known in terms of the density c and the pa-rameters of the Prigogine-Herman kinetic model. This result is perhaps somewhat surprising, as one might reasonably have expected rather a continuum higher-order model of the type suggested by Payne (1971). Figure 11.2 illustrates how an initial discontinuity between an upstream higher-density re-gion and a downstream lower-density region will tend to dis-sipate according to the diffusively corrected Lighthill-Whitham model, as opposed to the shock wave predicted by ordinary Lighthill-Whitham theory, which is given by Eq. (11.2) with D � 0. See Nelson (2000) for more details of the example underlying this figure. Sopasakis (2000) has also

developed the zero-order (again Lighthill-Whitham) and first-order Hilbert expansions, and the second-order Chapman-Enskog expansion, for the Prigogine-Herman kinetic model.

Interest in continuum models of traffic flow seems likely to continue, as applications exist within the space of resolution/fidelity/scale requirements for which continuum models are deemed most suitable. Along with this, interest in the use of kinetic models of vehicular traffic as a basis for continuum models seems likely to continue. For example, the venerable Lighthill-Whitham (1955) model is widely viewed as the most basic continuum model of traffic flow. But a suit-able traffic stream model is an essential ingredient of the Lighthill-Whitham model. Thus, traffic stream models are an important part of continuum models, as well as being of inter-est in their own right. Therefore, both of the benchmarks de-marcated in the preceding section can be viewed as related to the issue of how well a particular kinetic model performs in terms of providing a particularly low-order continuum model, specifically the Lighthill-Whitham model.

11.5 Direct Solution of Kinetic Equations Along with the traditional application of kinetic models of vehicular traffic to rational development of continuum models from microscopic models, as described in the preceding sec-tion, modern computers permit consideration of the utility of kinetic models in their own right, rather than merely as tools that can be used to construct continuum models. In this re-spect, there are two substantive issues: i) How can one solve kinetic equations, to obtain the distri-

bution function (f)? ii) Given this distribution function, what applications of it

can usefully be made? As regards the first issue, Hoogendoorn and Bovy (to appear) have employed Monte Carlo (i.e., simulation-based) tech-niques for the computational solution of a kinetic equation of vehicular traffic that builds upon the earlier work of Paveri-Fontana (1975). By contrast, in the kinetic theory of gases there exists a significant body of knowledge (e.g., Neunzert and Struckmeier, 1995, and other works cited therein) relative to the deterministic computational solution of kinetic equa-tions. This knowledge base undoubtedly could be invaluable in attempting to develop similar capabilities for vehicular traf-fic, but the equations are sufficiently different from those aris-ing in the kinetic theory of gases so that considerable further development is likely to be necessary. This

KINETIC THEORIES

11-8

Figure 11-2 Evolution of the flow, according to a diffusively corrected Lighthill-Whitham model, from initial conditions con-sisting of 190 vpm downstream of x=0 and 30 vpm upstream. See Nelson (2000) for details of the traffic stream model (flow function) and diffusion coefficient.

development is unlikely to occur in the absence of a relatively clear vision as to the uses that would be made of it. Therefore, the focus here primarily will be on the second of these issues.

Any consideration of applications of the distribution function of a kinetic theory requires a consideration of its in-terpretation. It is a statistical distribution function. The tradi-tional interpretation of such a distribution function is that it describes the frequency with which certain properties occur among samples drawn from some sample space. In the kinetic theory of vehicular traffic, the samples are vehicles, and the properties of interest are the positions and speeds of the vehi-cles; however, there are two fundamentally different possible interpretations of the underlying sample space:

1. Single-instance sampling: The sample space consists of all vehicles present on a specified road network at a spe-cific designated time.

2. Ensemble sampling: The sample space consists of all vehicles present, at a designated time, on one of an en-semble of identical road networks.

For example, the Houston freeway network at 5:00 p.m. on Wednesday, July 15, 1998 would be a reasonable sample space for single-instance sampling. On the other hand, the Houston freeway network at 5:00 p.m. on all midweek work-days during 1998 for which dry weather conditions prevailed would be a reasonable approximation of a sample space suit-able for ensemble sampling.

The difference between these two interpretations is subtle, but it has profound consequences. Traffic theorists

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0

0.1

0.2

0.3

0

50

100

150

200

250

distance (miles)

time (hours)

dens

ity (v

ehic

les

per m

ile)

KINETIC THEORIES

11-9

normally tend to think in terms that are most consistent with single-instance sampling. But any attempt to apply the kinetic theory within that interpretation implies the intention to pre-dict, at some level of approximation, the evolution of traffic for that specific instance, given suitable initial and boundary conditions for the distribution function. It seems somewhat questionable that this is attainable over any significant dura-tion. (The ``rolling horizon’’ approach often applied to pre-diction of traffic flow is a tacit admission of the significance of this issue.) On the other hand, the ensemble sampling in-terpretation implies only the intent to predict the likelihood with which various outcomes will occur. This intuitively seems much more achievable (cf. p. 10 of Asimov, 1988). Thus the more subtle ensemble sampling interpretation leads to an apparently more achievable objective than does the more obvious single-instance interpretation. For that reason, the ensemble sampling interpretation seems more likely to lead to potential direct applications of the kinetic theory.

The fundamental advantage of kinetic models over continuum models is that the kinetic distribution function pro-vides an estimate of the variability (over various instances of the ensemble, under the ensemble-sampling interpretation) of densities and speed at specific times and locations, whereas continuum models provide estimates of only the mean (pre-sumably over the ensemble) of these quantities. If the quantity (function of position and time) of interest in a particular appli-cation is not highly variable between instances within the en-semble, or if that variability is not of interest, then presumably one should choose a continuum model, or perhaps an even more highly aggregated model. On the other hand, if this variability is both of significant magnitude and important to the issue under study, then kinetic models might be a useful alternative to the computationally more expensive possibility of running a sufficiently large number of microscopic simula-tions so as to capture the nature of the variation between in-stances.

Some specific instances of quantities for which vari-ability might be of significant interest are travel times and driving cycles. The latter require knowledge about the statis-tical distribution of accelerations, as well as velocities and densities, but such acceleration information is inherent in the distribution function, along with the mechanical and correla-tion models that underlie any kinetic model. In fact, kinetic models seem to be the natural connecting link between contin-uum models, which provide the “cross sectional” view of traffic that most transportation planning is based on, and the “longitudinal” view that underlies the standard driving cycle

approach to estimation of fuel emissions (cf. Carson and Aus-tin, 1997).

The crucial question underlying any potential appli-cation of kinetic models is whether a kinetic model can be found that has sufficient fidelity and resolution for the particu-lar application, and that can be solved on the necessary scale using available computational resources. The answer to that clearly depends upon the specific details of the particular ap-plication, and any such proposed application of a specific ki-netic model must be validated against actual observations. However, data of sufficiently high quality to permit such vali-dations are both rare and expensive to obtain. Under these circumstances, it seems appropriate to use microscopic models (e.g., cellular automata) as a framework within which initially to vet proposed kinetic models.

Specifically, it seems worthwhile to employ micro-scopic models to study the following: HYPOTHESIS: The multiparameter family of equilibrium solu-tions of the Prigogine-Herman kinetic model found by Nelson and Sopasakis (1998), with its attendant traffic stream surface (rather than the classical curve), reflects the fact that actual traffic has a number of spatially homogeneous equilibrium states (with different average speeds) corresponding to the same density. If this hypothesis is true, then presumably different initial con-figurations of a traffic stream have the possibility to approach different states in the long-time limit, even though their densi-ties are the same on the macroscopic scale. Results reported by Nagel (1996, esp. Sec. V) tend to confirm this hypothesis.

References Asimov, I. (1988). Prelude to Foundation. Doubleday, New

York. Beylich, A. E. (1979). Elements of a Kinetic Theory of Traffic

Flow, Proceedings of the Eleventh International Sympo-sium on Rarefied Gas Dynamics, Commissariat a L’Energie Atomique, Paris, pp. 129-138.

Bui, D. D., P. Nelson and A. Sopasakis (1996). The General-ized Bimodal Traffic Stream Model and Two-regime Flow Theory. Transportation and Traffic Theory, Proceedings of the 13th International Symposium on Transportation and Traffic Theory, Pergamon Press (Jean-Baptiste Le-sort, Ed.), Oxford, pp. 679-696.

Carson, T. R. and T. C. Austin, Development of Speed Cor-rection Cycles. Report prepared for the U. S. Environ-mental Protection Agency, Contract No. 68-C4-0056,

KINETIC THEORIES

11-10

Work Assignment No. 2-01, Sierra Research, Inc., Sac-ramento, California, April 30..

Ceder, A. (1976). A Deterministic Flow Model of the Two-regime Approach. Transportation Research Record 567,TRB, NRC, Washington, D.C., pp. 16-30..

Disbro, J. E. and M. Frame (1989). Traffic Flow Theory and Chaotic Behavior. Transportation Research Record 1225,TRB, NRC, Washington, D.C., pp. 109-115.

Drake, J. S., J. L. Schofer and A. D. May, Jr. (1967). A Statis-tical Analysis of Speed Density Hypothesis. Highway Re-search Record 154, TRB. NRC, Washington, D.C., pp. 53-87.

Edie, L. C., R. Herman and T. N. Lam (1980). Observed Mul-tilane Speed Distribution and the Kinetic Theory of Ve-hicular Traffic. Transportation Research Volume 9, pp. 225-235,

Grad, H. (1958). Principles of the Kinetic Theory of Gases.Handbuch der Physik, Vol. XII, Springer-Verlag, Berlin.

Hall, F. L. (1987). An Interpretation of Speed-Flow Concen-tration Relationships Using Catastrophe Theory. Trans-portation Research A, Volume 21, pp. 191-201.

Helbing, D. (1995). High-fidelity Macroscopic Traffic Equa-tions. Physica A, Volume 219, pp. 391-407.

Helbing, D., “Gas-Kinetic Derivation of Navier-Stokes-like Traffic Equations,” Physical Review E. 53, 2366-2381, 1996.

Hoogendoorn, S.P., and P.H.L. Bovy (to appear). Non-Local Multiclass Gas-Kinetic Modeling of Multilane Traffic Flow. Networks and Spatial Theory.

Kerner, B. S. and P. Konhäuser (1993). Cluster Effect in Ini-tially Homogeneous Traffic Flow. Physical Review E, Volume 48, pp. R2335-R2338.

Klar, A. and R. Wegener (1999). A Hierarchy of Models for Multilane Vehicular Traffic (Part I: Modeling and Part II: Numerical and Stochastic Investigations). SIAM J. Appl. Math, Volume 59, pp. 983-1011.

Lighthill, M. J. and G. B. Whitham (1955). On Kinematic Waves II: A Theory of Traffic Flow on Long Crowded Roads. Proceedings of the Royal Society, Series A, Vol-ume 229, pp. 317-345.

Nagel, K. (1996). Particle Hopping Models and Traffic Flow Theory. Physical Review E, Volume 53, pp. 4655-4672.

Nagel, K. (1998). From Particle Hopping Models to Traffic Flow Theory. Transportation Research Record, Volume 1644, pp. 1-9.

Munjal, P. and J. Pahl (1969). An Analysis of the Boltzmann-type Statistical Models for Multi-lane Traffic Flow.Transportation Research, Volume 3, pp. 151-163.

Nelson, P. (1995a). A Kinetic Model of Vehicular Traffic and its Associated Bimodal Equilibrium Solutions. Transport Theory and Statistical Physics, Volume 24, pp. 383-409.

Nelson, P. (1995b). On Deterministic Developments of Traffic Stream Models. Transportation Research B, Volume 29, pp. 297-302.

Nelson, P. (2000). Synchronized Flow from Modified Lighthill-Whitham Model. Physical Review E, Volume 61, pp. R6052-R6055.

Nelson, P., D. D. Bui and A. Sopasakis (1997). A Novel Traf-fic Stream Model Deriving from a Bimodal Kinetic Equi-librium. Preprints of the IFAC/IFIP/IFORS Symposium on Transportation Systems, Technical University of Crete (M. Papageorgiou and A. Pouliezos, Eds.), Volume 2, pp. 799-804.

Nelson, P. and A. Sopasakis (1998). The Prigogine-Herman Kinetic Model Predicts Widely Scattered Traffic Flow Data at High Concentrations. Transportation Research B, Volume 32, pp. 589-604.

Nelson, P. and A. Sopasakis (1999). The Chapman-Enskog Expansion: A Novel Approach to Hierarchical Extension of Lighthill-Whitham Models. Transportation and Traffic Theory: Proceedings of the 14th International Symposium on Transportation and Traffic Theory, Pergamon (A. Ceder, Ed.), pp. 51-80.

Neunzert, H. and J. Struckmeier (1995). Particle Methods for the Boltzmann Equation. Acta Numerica, Volume 4, pp. 417-457.

Paveri-Fontana, S. L. (1975). On Boltzmann-like Treatments for Traffic Flow: A Critical Review of the Basic Model and an Alternative Proposal for Dilute Traffic Analysis.Transportation Research, Volume 9, pp. 225-235.

Payne H. J. (1971). Models of Freeway Traffic and Control. Simulation Council Procs., Simulations Council, Inc., La Jolla, CA, Vol. 1, pp. 51-61.

Phillips, W. F. (1977). Kinetic Model for Traffic Flow. Re-port DOT/RSPD/DPB/50-77/17, U. S. Department of Transportation.

Phillips, W. F. (1979). A Kinetic Model for Traffic Flow with Continuum Implications. Transportation Planning and Technology, Volume 5, pp. 131-138.

Prigogine I. and F. C. Andrews (1960). A Boltzmann-like Ap-proach for Traffic Flow. Operations Research, Volume 8, pp. 789-797.

Prigogine, I. and R. Hermann (1970). Comment on “An Analysis of the Boltzmann-type Statistical Models for Multi-lane Traffic Flow.” Transportation Research, Vol-ume 4, pp. 113-116.

KINETIC THEORIES

11-11

Prigogine, I. and R. Hermann (1971). Kinetic Theory of Ve-hicular Traffic, Elsevier, New York.

Sopasakis, A. (2000). Developments in the Theory of the Prigogine-Herman Kinetic Equation of Vehicular Traffic. Ph.D. dissertation (mathematics), Texas A&M University, May.

INDEX

Note: Index has not been updated to reflect revisions to Chapters 2, 5 and new 11.

1 2 - 1

A B

AASHTO Green Book, 3-21 ballistic, 3-8acceleration of the lead car, fluctuation in the, 4-8 bifurcation behavior, 5-15acceleration control, 3-24 block length, average, 6-20, 6-20, 6-22acceleration noise, 7-8 blockages per hour, 6-22actuated signals , 9-23 boundary, 5-4, 5-4, 5-10-5-11, 5-23, 5-24, 5-36adaptive signals, 9-19 brake and carburetion systems, 7-8adaptive signal control, 9-27 braking inputs, 3-7aerial photography, 2-3, 6-11 braking performances, 3-20, 4-1aerodynamic conditions, 7-9 braking performance reaction time, 3-5aero-dynamic effects, 7-11age, 3-16aggregated data , 6-3aggressive driving, 6-20aging eyes, 3-16air pollutant levels, 7-15air pollutants, 7-13, 7-14air pollution, 7-13air quality standards, 7-14, 7-15air quality models, 7-15air quality, 7-13, 7-15air resistance, 7-12�-relationship, 6-12alternative fuel, 7-15alternative fuels, 7-12altitude, 7-8, 7-8ambient temperature, 7-8, 7-8Ambient Air Quality Standards, 7-13analytical solution, 5-3, 5-3, 5-9arrival and departure patterns, 5-9, 5-9arterials, 5-6, 5-9 Athol, 2-2, 2-10, 2-22auxiliary electric devices, 7-8average block length, 6-20, 6-20, 6-22average cycle length, 6-20average flows, 6-8average maximum running speed, 6-17average number of lanes per street, 6-20, 6-22average road width, 6-6average signal cycle length, 6-20, 6-23average signal spacing, 6-10average space headway, 5-6, 5-6average speed,6-3, 6-6, 6-8, 6-10, 6-11, 6-17, 6-22,

7-9, 7-11average speed limit, 6-20average street width, 6-10, 6-11

C

California standards, 7-15CALINE-4 dispersion model, 7-15capacity, 4-1carbon monoxide, 7-13car-following,10-2, 10-3, 10-8, 10-15car following models, 4-1catastrophe theory, 2-8, 2-27, 2-28central city, 6-8central vs. peripheral processes, 3-17changeable message signs, 3-12changes in cognitive performance, 3-17changes in visual perception, 3-16chase car, 6-21, 6-22Clean Air Act, 7-13, 7-13closed-loop braking performance, 3-21coefficient of variation, 3-11cognitive changes, 3-16collective flow regime, 6-16composite emission factors, 7-15compressibility, 5-1, 5-9compressible gases, 5-22, 5-22computer simulation, 6-22, 6-23concentration, 2-1, 2-5, 2-8, 2-20, 2-29, 4-15, 6-16,

6-17, 6-20, 6-23concentration at maximum flow, 6-25conditions, 5-4, 5-6-5-9, 5-11, 5-23, 5-27, 5-29-5-30,

5-32, 5-36, 5-38, 5-43, 5-45confidence intervals, 10-17, 10-17, 10-20, 10-21,

10-26congested operations, 2-11, 2-22continuity equation, 5-1-5-3, 5-20, 5-22, 5-24, 5-25continuous simulation models, 10-3continuum models, 5-1-5-1, 5-3, 5-20, 5-29, 5-41control, 3-1, 4-2control movement time, 3-7, 3-7control strategies, 6-22

1 2 - 2

convection motion and relaxation, 5-20 entrance or exit ramps, 5-12convection term, 5-20, 5-22 equilibrium, 5-1-5-1, 5-3, 5-10, 5-22-5-23, 5-45convergence, 5-11 ergodic, 6-17coordinate transformation method., 9-11 evasive maneuvers, 3-15correlation methods, 10-22, 10-22 expectancies, 3-7critical gap values for unsignalized exposure time, 3-13

intersections, 3-26cruising, 7-11 cruising speed, 7-11

D

deceleration-acceleration cycle, 7-11, 7-11decision making, 4-2defensive driving, 3-17-3-17delay models at isolated signals, 9-2delay per intersection, 6-10density, 1-4-1-4, 2-1- 2-3, 2-7, 2-11, 2-18, 2-21, 2-28density and speed, 2-3, 2-22disabled drivers, 3-2discontinuity, 5-1, 5-4discrete simulation models, 10-3discretization, 5-10, 5-10-5-14, 5-26, 5-30, 5-34display for the driver, 3-2dissipation times, 5-8distractors on/near roadway, 3-28disturbance, 4-15Drake et al., 2-7, 2-12, 2-20, 2-23, 2-24, 2-28, 2-36driver as system manager, 3-2driver characteristics, 7-8driver performance characteristics, 3-28driver response or lag to changing traffic signals, 3-9drivers age, 3-16driving task, 3-9, 3-28drugs, 3-17-3-17

E

Edie, 2-6, 2-18, 2-32, 2-34effective green interval, 5-6, 5-8effective red interval, 5-8electrification, 7-15Elemental Model, 7-9, 7-11emission control, 7-14emissions, 7-13energy consumption, 7-8, 7-12energy savings, 7-8engine size, 7-8, 7-12engine temperature, 7-8

F

figure/ground discrimination, 3-17, 3-17filtering effect on signal performance, 9-17first and second moments, 5-22Fitts' Law, 3-7, 3-7fixed-time signals , 9-23floating car procedure, 2-3floating vehicles, 6-11flow, 1-4, 1-4, 2-1, 2-7, 2-10, 2-16, 2-18, 2-24, 2-26,

2-32, 2-34, 4-1flow-concentration relationship, 4-15flow rates, 2-2, 2-4-2-5, 2-14, 2-32, 5-3, 5-6, 5-10,

5-12, 5-13, 5-19, 5-24, 5-25forced pacing under highway conditions, 3-17fraction of approaches with signal progression, 6-20fraction of curb miles with parking, 6-20, 6-23fraction of one-way streets, 6-20, 6-20, 6-22fraction of signalized approaches in progression, 6-23fraction of signals actuated, 6-20fraction of vehicles stopped, 6-17, 6-23free-flow speed, 6-9fuel consumption, 6-23, 7-8, 7-9, 7-12fuel consumption models, 7-8fuel consumption rate, 7-8, 7-9, 7-11fuel efficiency, 7-8, 7-9, 7-12fundamental equation, 2-8, 2-10

G

gap acceptance, 3-10, 3-25gasoline type, 7-8gasoline volatility, 7-15Gaussian diffusion equation, 7-15gender, 3-16, 3-16glare recovery, 3-17“good driving" rules, 4-1grades, 7-8Greenberg, 2-20, 2-20, 2-21, 2-34Greenshields, 2-18, 2-18, 2-34guidance, 3-1

1 2 - 3

H

headways, 2-2, 2-2, 2-3, 2-8Hick-Hyman Law, 3-3high order models, 5-1-5-1, 5-15Highway Capacity Manual, 4-1highway driving, 7-11Human Error, 3-1humidity, 7-15hysteresis phenomena, 5-15

I

identification, 3-9, 3-15idle flow rate, 7-12idling, 7-11, 7-11Index of Difficulty, 3-8individual differences in driver performance, 3-16infinitesimal disturbances, 4-15 May, 2-2-2-7, 2-9, 2-12, 2-22, 2-24, 2-33, 2-36information filtering mechanisms, 3-17 measurements along a length of road, 2-3information processor, 3-2 Measures of Effectiveness, 10-17, 10-17, 10-25initial and boundary conditions, 5-5, 5-5, 5-6, 5-11 medical conditions, 3-18inner zone, 6-10 merging, 3-25inspection and maintenance, 7-15 meteorological data, 7-15instantaneous speeds, 7-12 methanol, 7-15interaction time lag, 5-12, 5-12, 5-13 microscopic, 6-22intersection capacity, 6-11 microscopic analyses , 6-1intersection density, 6-20 minimum fraction of vehicles stopped, 6-25intersections per square mile, 6-20 minimum trip time per unit distance, 6-17, 6-17 mixingintersection sight distance, 3-10, 3-27 zone, 7-16Intelligent Transportation Systems (ITS), method of characteristics, 5-4

2-1-2-2, 2-5, 2-6, 2-8, 2-19-2-20, 2-24, model validation, 10-52-32-2-33, 3-1, 6-25 model verification, 10-5, 10-15

J

jam concentration, 4-14jam density, 5-3, 5-8, 5-11-5-14

K

kinetic theory of traffic flow, 6-16

L

lane-changing, 10-5lane miles per square mile, 6-20

lead (Pb), 7-13legibility , 3-9levels of service, 6-2light losses and scattering in optic train, 3-16local acceleration, 5-20, 5-26log-normal probability density function, 3-5looming, 3-13loss of visual acuity, 3-16

M

macroscopic, 6-1macroscopic measure, 6-16macroscopic models, 6-6macroscopic relations, 6-25macular vision, 3-17maximum average speed, 6-3maximum flow, 6-11

momentum equation, 5-1-5-1, 5-22, 5-26, 5-29motion detection in peripheral vision, 3-14movement time, 3-7moving observer method, 2-3, 2-3MULTSIM, 7-12

N

navigation, 3-1NETSIM , 6-22, 6-23network capacity, 6-6network topology, 6-1network concentrations, 6-22, 6-24network features, 6-20, 6-20network-level relationships, 6-23network-level variables, 6-25

1 2 - 4

network model, 6-1, 6-6 period of measurement, 7-15network performance, 6-1 "Plain Old Driving" (POD), 3-1network types, 6-6 platoon dynamics, 5-6network-wide average speed, 6-8 platooning effect on signal performance, 9-15nighttime static visual acuity, 3-11 pollutant dispersion, 7-16nitrogen dioxide, 7-13non-instantaneous adaption, 5-23non-linear models, 4-15normal or gaussian distribution, 3-5normalized concentration, 4-15normalized flow, 4-15number of lanes per street, 6-20number of stops, 6-23, 7-8numerical solution, 5-9, 5-11, 5-12, 5-29, 5-31-5-33,

5-49

O

object detection, 3-15obstacle and hazard detection, 3-15obstacle and hazard recognition, 3-15obstacle and hazard identification , 3-15occupancy, 1-4, 2-1, 2-9, 2-11, 2-21, 2-22,

2-25-2-26, 2-28, 2-32, 2-34, 2-36off-peak conditions, 6-6Ohno's algorithm, 9-8oil viscosity, 7-8oncoming collision, 3-13open-loop, 3-8open-loop braking performance, 3-20oscillatory solutions, 5-15outer zone, 6-10overtaking and passing in the traffic stream, 3-24overtaking and passing vehicles, 3-24overtaking and passing vehicles (Opposing Traffic),

3-25oxygenated fuels/reformulated gasoline, 7-15ozone, 7-13

P

partial differential equation, 5-4, 5-30particulate matter, 7-13pavement roughness, 7-8pavement type, 7-8peak conditions, 6-6perception-response time, 3-3peripheral vs. central processes, 3-17perception, 4-2

Positive Guidance, 3-28positive kinetic energy, 7-11pupil, 3-16

Q

quality of service, 6-20, 6-25quality of traffic service, 6-12, 6-16queue, 5-4, 5-7, 5-50queue discharge flow, 2-12, 2-13, 2-15queue length, 5-6, 5-9queue length stability, 5-8

R

radial motion, 3-13random numbers, 10-2-10-2, 10-22, 10-26reaction time, 3-3, 3-3, 3-4, 3-7, 3-8, 3-16, 3-17real-time driver information input, 3-28refueling emissions controls, 7-15relaxation term, 5-23resolving power, 3-11response distances and times to traffic control

devices, 3-9response time, 3-4, 3-7, 3-15, 3-16, 3-20response to other vehicle dynamics, 3-13road density, 6-15roadway gradient, 7-8rolling friction, 7-9rolling resistance, 7-12running (moving) time, 6-17running speed, 6-10, 6-10, 6-11

S

saturation flow, 6-10scatter in the optic train, 3-17scattering effect of, 3-17senile myosis, 3-16sensitivity coefficient , 4-15, 5-12, 5-12shock waves, 5-1, 5-1, 5-3-5-4, 5-6, 5-29, 5-30, 5-50signalized intersection, 5-6, 5-6, 5-7signalized links and platoon behavior, 5-9

1 2 - 5

short-term events, 6-22 stimulus-response equation, 4-3signals, stochastic process, 10-17

actuated, 9-23 stochastic simulation, 10-5adaptive, 9-19 stop time, 6-17, 6-17

signal control,adaptive, 9-27 stopped time, 6-10signal densities, 6-10 stopped delay, 7-11signal density, 6-20 stopping maneuvers, 3-15signals per intersection, 6-22 stopping sight distance, 3-26sign visibility and legibility, 3-11 stop-start waves, 5-15-5-15, 5-17, 5-24, 5-26, 5-36, signage or delineation, 3-17 5-39simulation models,building 10-5 street network, 6-20site types, 7-15 structure chart, 10-8smog, 7-13 substantial acceleration, 5-20, 5-20Snellen eye chart, 3-11 sulfur dioxide, 7-13sound velocity, 5-22 summer exodus to holiday resorts, 5-17source emissions, 7-14 surface conditions, 7-8space headway, 2-1, 2-5 suspended particulate , 7-13space mean speed, 2-6-2-7, 2-9-2-10, 6-15, 7-11spacing, 2-1, 2-1, 2-26, 4-8, 5-2, 5-17, 5-29, 5-34specific maneuvers at the guidance level, 3-24speed, 2-3, 2-6, 2-8, 2-11, 2-14, 2-16, 2-18, 2-22,

2-24, 2-28, 2-31, 2-33, 4-1, 4-15speed (miles/hour) versus vehicle concentration

(vehicles/mi), 4-17speed and acceleration performance, 3-24speed-concentration relation, 4-13speed-density models, 2-19speed-density relation, 5-15-5-15, 5-20, 5-22-5-23,

5-27, 5-34speed-flow models,2-13, 2-1,9 6-8speed-flow relation, 6-6speeds from flow and occupancy, 2-8, 2-9speed limit changes, 3-28speed noise, 7-8, 7-12speed of the shock wave, 5-4speed-spacing, 4-15speed-spacing relation, 4-1spillbacks, 5-9stability analysis, 5-8, 5-25, 5-28-5-29, 5-43standard deviation of the vehicular speed distribution,

5-22, 5-39state equations, 5-9, 5-9State Implementation Plans (SIPs), 7-15stationary sources, 7-13statistical distributions, 10-5, 10-6steady-state, 7-11steady-state delay models, 9-3steady-state expected deceleration, percentile estimatesof , 3-21steady-state flow, 4-15steady-state traffic speed control, 3-24steering response times, 3-9, 3-9

T

tail end, 5-6-5-8temperature, 7-15time-dependent delay models, 9-10time headway, 2-1time mean speed, 2-6-2-7tire pressure, 7-8tire type, 7-8total delay, 6-23total trip time, 6-17TRAF-NETSIM , 6-22, 6-23traffic breakdowns, 5-15, 5-42traffic conditions, 7-9traffic control devices (TCD), 3-9traffic control system, 6-1traffic data, 7-15traffic dynamic pressure, 5-23traffic intensity , 6-2, 6-15traffic network, 6-1traffic performance, 6-1traffic signal change, 3-9traffic simulation, 10-1-10-2, 10-4, 10-7, 10-15-

10-17, 10-20, 10-22 traffic stream, 4-1 trajectories of vehicles, 5-4trajectory, 5-4, 5-7-5-9transients, 5-15-5-15, 5-17, 5-20transmission type, 7-8travel demand levels, 6-1travel time , 6-1, 6-10trip time per unit distance, 7-9two-fluid model , 6-1, 6-17, 6-22-6-23, 6-25

1 2 - 6

two-fluid parameters, 6-18, 6-18, 6-20, 6-23, 6-25 vehicle miles traveled, 6-11two-fluid studies, 6-20 vehicle shape, 7-8two-fluid theory, 6-12, 6-16, 6-24 vehicles stopped ,average fraction of the, 6-17turning lanes, 5-9, 5-9 viscosity, 5-22, 5-24, 5-29, 5-34

U

undersaturation, 5-8effect of upstream signals, 9-15UMTA, 7-15UMTA Model, 7-15urban driving cycle, 7-11urban roadway section, 7-11uncongested flows, 2-12

V

variability among people, 3-16vehicle ahead, 3-13vehicle alongside, 3-14vehicle characteristics, 7-8vehicle emissions, 7-14vehicle fleet, 7-8vehicle mass, 7-8, 7-9, 7-12

visual acuity, 3-11visual angle, 3-11-3-13, 3-15, 3-16visual performance, 3-11volatile organic compounds, 7-13

W

Wardrop, 2-4, 2-4, 2-6-2-7Wardrop and Charlesworth, 2-4, 2-4Weber fraction, 3-13, 3-13wheel alignment, 7-8wind, 7-8wind conditions, 7-8wind speed, 7-15work zone traffic control devices, 3-17

Y

yield control for secondary roadway, 3-27

traffic flow theory 2001

Documents